2.3.3 Problem 3

Internal problem ID [18468]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 29. Problems at page 81
Problem number : 3
Date solved : Thursday, March 13, 2025 at 12:03:03 PM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

Solve

$$\begin{array}{rl}{y}^{\prime }& =-\frac{2ax+by-g}{2cy+bx+e}\end{array}$$

Solved as first order polynomial type ode

Time used: 0.411 (sec)

This is ODE of type polynomial. Where the RHS of the ode is ratio of equations of two lines. Writing the ODE in the form

$$y^{\prime }=\frac{a_{1}x+b_{1}y+c_1}{a_{2}x+b_{2}y+c_3}$$

Where $a_1=-2a,b_1=-b,c_1=g,a_2=b,b_2=2c,c_2=e$. There are now two possible solution methods. The first case is when the two lines $a_{1}x+b_{1}y+c_1$,$a_{2}x+b_{2}y+c_3$ are not parallel and the second case is if they are parallel. If they are not parallel, then the transformation $X=x-x_0$, $Y=y-y_0$ converts the ODE to a homogeneous ODE. The values $x_0,y_0$ have to be determined. If they are parallel then a transformation $U(x)=a_{1}x+b_{1}y$ converts the given ODE in $y$ to a separable ODE in $U(x)$. The first case is when $\frac{a_1}{b_1}\ne \frac{a_2}{b_2}$ and the second case when $\frac{a_1}{b_1}=\frac{a_2}{b_2}$. From the above we see that $\frac{a_1}{b_1}\ne \frac{a_2}{b_2}$. Hence this is case one where lines are not parallel. Using the transformation

$$\begin{array}{rl}X& =x-x_0\\ Y& =y-y_0\end{array}$$

Where the constants $x_0,y_0$ are obtained by solving the following two linear algebraic equations

$$\begin{array}{rl}{a}_1{x}_0+b_1{y}_0+c_1& =0\\ a_2{x}_0+b_2{y}_0+c_2& =0\end{array}$$

Substituting the values for $a_1,b_1,c_1,a_2,b_2,c_2$ gives

$$\begin{array}{rl}-2a{x}_0-b{y}_0+g& =0\\ b{x}_0+2c{y}_0+e& =0\end{array}$$

Solving for $x_0,y_0$ from the above gives

$$\begin{array}{rl}{x}_0& =\frac{be+2cg}{4ac-b^2}\\ y_0& =-\frac{2ae+bg}{4ac-b^2}\end{array}$$

Therefore the transformation becomes

$$\begin{array}{rl}X& =x-|\frac{be+2cg}{4ac-b^2}|\\ Y& =y+|\frac{2ae+bg}{4ac-b^2}|\end{array}$$

Using this transformation in $2ax+by+(2cy+bx+e)y^{\prime }=g$ result in

$$\begin{array}{rl}\frac{dY}{dX}& =\frac{-2Xa-Yb}{Xb+2Yc}\end{array}$$

This is now a homogeneous ODE which will now be solved for $Y(X)$. In canonical form, the ODE is

$$\begin{array}{rl}{Y}^{\prime }& =F(X,Y)\\ \text{(1)}& & =-\frac{2Xa+Yb}{Xb+2Yc}\end{array}$$

An ode of the form $Y^{\prime }=\frac{M(X,Y)}{N(X,Y)}$ is called homogeneous if the functions $M(X,Y)$ and $N(X,Y)$ are both homogeneous functions and of the same order. Recall that a function $f(X,Y)$ is homogeneous of order $n$ if

$$f(t^{n}X,t^{n}Y)=t^{n}f(X,Y)$$

In this case, it can be seen that both $M=-2Xa-Yb$ and $N=Xb+2Yc$ are both homogeneous and of the same order $n=1$. Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution $u=\frac{Y}{X}$, or $Y=uX$. Hence

$$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{\mathrm{d}u}{\mathrm{d}X}X+u$$

Applying the transformation $Y=uX$ to the above ODE in (1) gives

$$\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}X}X+u& =\frac{-bu-2a}{2cu+b}\\ \frac{\mathrm{d}u}{\mathrm{d}X}& =\frac{\frac{-bu\left(X\right)-2a}{2cu\left(X\right)+b}-u\left(X\right)}{X}\end{array}$$

Or

$$\frac{d}{dX}u\left(X\right)-\frac{\frac{-bu\left(X\right)-2a}{2cu\left(X\right)+b}-u\left(X\right)}{X}=0$$

Or

$$2u\left(X\right)\left(\frac{d}{dX}u\left(X\right)\right)Xc+2u{\left(X\right)}^{2}c+\left(\frac{d}{dX}u\left(X\right)\right)Xb+2bu\left(X\right)+2a=0$$

Or

$$X(2cu\left(X\right)+b)\left(\frac{d}{dX}u\left(X\right)\right)+2u{\left(X\right)}^{2}c+2bu\left(X\right)+2a=0$$

Which is now solved as separable in $u\left(X\right)$.

The ode

$$\begin{array}{}\text{(1)}& \frac{d}{dX}u\left(X\right)=-\frac{2(u{\left(X\right)}^{2}c+bu\left(X\right)+a)}{X(2cu\left(X\right)+b)}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}\frac{d}{dX}u\left(X\right)& =-\frac{2(u{\left(X\right)}^{2}c+bu\left(X\right)+a)}{X(2cu\left(X\right)+b)}\\ & =f(X)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(X)& =-\frac{2}{X}\\ g(u)& =\frac{u^{2}c+bu+a}{2cu+b}\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(X)dX\\ \int \frac{2cu+b}{u^{2}c+bu+a}du& =\int -\frac{2}{X}dX\end{array}$$

$$\ln (u{\left(X\right)}^{2}c+bu\left(X\right)+a)=\ln \left(\frac{1}{X^2}\right)+c_2$$

Taking the exponential of both sides the solution becomes

$$u{\left(X\right)}^{2}c+bu\left(X\right)+a=\frac{c_2}{X^2}$$

Converting $u{\left(X\right)}^{2}c+bu\left(X\right)+a=\frac{c_2}{X^2}$ back to $Y\left(X\right)$ gives

$$\begin{array}{r}Y{\left(X\right)}^{2}c+bY\left(X\right)X+a{X}^2=c_2\end{array}$$

The solution is implicit $Y{\left(X\right)}^{2}c+bY\left(X\right)X+a{X}^2=c_2$. Replacing $Y=y-y_0,X=x-x_0$ gives

$$a{(x-\frac{be+2cg}{4ac-b^2})}^2+(y+\frac{2ae+bg}{4ac-b^2})b(x-\frac{be+2cg}{4ac-b^2})+{(y+\frac{2ae+bg}{4ac-b^2})}^{2}c=c_2$$

Summary of solutions found

$$\begin{array}{rl}a{(x-\frac{be+2cg}{4ac-b^2})}^2+(y+\frac{2ae+bg}{4ac-b^2})b(x-\frac{be+2cg}{4ac-b^2})+{(y+\frac{2ae+bg}{4ac-b^2})}^{2}c& =c_2\end{array}$$

Solved as first order homogeneous class Maple C ode

Time used: 0.930 (sec)

Let $Y=y-y_0$ and $X=x-x_0$ then the above is transformed to new ode in $Y(X)$

$$\frac{d}{dX}Y\left(X\right)=-\frac{2a(X+x_0)+b(Y\left(X\right)+y_0)-g}{2c(Y\left(X\right)+y_0)+b(X+x_0)+e}$$

Solving for possible values of $x_0$ and $y_0$ which makes the above ode a homogeneous ode results in

$$\begin{array}{rl}{x}_0& =\frac{be+2cg}{4ac-b^2}\\ y_0& =\frac{-2ae-bg}{4ac-b^2}\end{array}$$

Using these values now it is possible to easily solve for $Y\left(X\right)$. The above ode now becomes

$$\begin{array}{r}\frac{d}{dX}Y\left(X\right)=-\frac{2aX+bY\left(X\right)+\frac{2a(be+2cg)}{4ac-b^2}+\frac{b(-2ae-bg)}{4ac-b^2}-g}{bX+2cY\left(X\right)+\frac{b(be+2cg)}{4ac-b^2}+\frac{2c(-2ae-bg)}{4ac-b^2}+e}\end{array}$$

In canonical form, the ODE is

$$\begin{array}{rl}{Y}^{\prime }& =F(X,Y)\\ \text{(1)}& & =-\frac{2aX+bY}{bX+2cY}\end{array}$$

An ode of the form $Y^{\prime }=\frac{M(X,Y)}{N(X,Y)}$ is called homogeneous if the functions $M(X,Y)$ and $N(X,Y)$ are both homogeneous functions and of the same order. Recall that a function $f(X,Y)$ is homogeneous of order $n$ if

$$f(t^{n}X,t^{n}Y)=t^{n}f(X,Y)$$

In this case, it can be seen that both $M=-2aX-bY$ and $N=bX+2cY$ are both homogeneous and of the same order $n=1$. Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution $u=\frac{Y}{X}$, or $Y=uX$. Hence

$$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{\mathrm{d}u}{\mathrm{d}X}X+u$$

Applying the transformation $Y=uX$ to the above ODE in (1) gives

$$\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}X}X+u& =\frac{-bu-2a}{2cu+b}\\ \frac{\mathrm{d}u}{\mathrm{d}X}& =\frac{\frac{-bu\left(X\right)-2a}{2cu\left(X\right)+b}-u\left(X\right)}{X}\end{array}$$

Or

$$\frac{d}{dX}u\left(X\right)-\frac{\frac{-bu\left(X\right)-2a}{2cu\left(X\right)+b}-u\left(X\right)}{X}=0$$

Or

$$2\left(\frac{d}{dX}u\left(X\right)\right)u\left(X\right)Xc+\left(\frac{d}{dX}u\left(X\right)\right)Xb+2u{\left(X\right)}^{2}c+2bu\left(X\right)+2a=0$$

Or

$$X(2cu\left(X\right)+b)\left(\frac{d}{dX}u\left(X\right)\right)+2u{\left(X\right)}^{2}c+2bu\left(X\right)+2a=0$$

Which is now solved as separable in $u\left(X\right)$.

The ode

$$\begin{array}{}\text{(2)}& \frac{d}{dX}u\left(X\right)=-\frac{2(u{\left(X\right)}^{2}c+bu\left(X\right)+a)}{X(2cu\left(X\right)+b)}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}\frac{d}{dX}u\left(X\right)& =-\frac{2(u{\left(X\right)}^{2}c+bu\left(X\right)+a)}{X(2cu\left(X\right)+b)}\\ & =f(X)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(X)& =-\frac{2}{X}\\ g(u)& =\frac{u^{2}c+bu+a}{2cu+b}\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(X)dX\\ \int \frac{2cu+b}{u^{2}c+bu+a}du& =\int -\frac{2}{X}dX\end{array}$$

$$\ln (u{\left(X\right)}^{2}c+bu\left(X\right)+a)=\ln \left(\frac{1}{X^2}\right)+c_1$$

Taking the exponential of both sides the solution becomes

$$u{\left(X\right)}^{2}c+bu\left(X\right)+a=\frac{c_1}{X^2}$$

Converting $u{\left(X\right)}^{2}c+bu\left(X\right)+a=\frac{c_1}{X^2}$ back to $Y\left(X\right)$ gives

$$\begin{array}{r}Y{\left(X\right)}^{2}c+bY\left(X\right)X+X^{2}a=c_1\end{array}$$

Using the solution for $Y(X)$

$$\begin{array}{}\text{(A)}& Y{\left(X\right)}^{2}c+bY\left(X\right)X+X^{2}a=c_1\end{array}$$

And replacing back terms in the above solution using

$$\begin{array}{rl}Y& =y+y_0\\ X& =x_0+x\end{array}$$

Or

$$\begin{array}{rl}Y& =y+\frac{-2ae-bg}{4ac-b^2}\\ X& =x+\frac{be+2cg}{4ac-b^2}\end{array}$$

Then the solution in $y$ becomes using EQ (A)

$$\begin{array}{r}{(y-\frac{-2ae-bg}{4ac-b^2})}^{2}c+b(y-\frac{-2ae-bg}{4ac-b^2})(x-\frac{be+2cg}{4ac-b^2})+a{(x-\frac{be+2cg}{4ac-b^2})}^2=c_1\end{array}$$

Solving for $y$ gives

$$\begin{array}{rl}y& =\frac{-4abcx+b^{3}x-4ace+b^{2}e+\sqrt{-64a^3{c}^3{x}^2+48a^2{b}^2{c}^2{x}^2-12a{b}^{4}c{x}^2+b^6{x}^2+32a^{2}b{c}^{2}ex+64a^2{c}^{3}gx-16a{b}^{3}cex-32a{b}^2{c}^{2}gx+2b^{5}ex+4b^{4}cgx+64c_1{a}^2{c}^3-32c_{1}a{b}^2{c}^2+4c_1{b}^{4}c-4a{b}^{2}c{e}^2-16ab{c}^{2}eg-16a{c}^3{g}^2+b^4{e}^2+4b^{3}ceg+4b^2{c}^2{g}^2}}{2c(4ac-b^2)}\\ y& =-\frac{4abcx-b^{3}x+4ace-b^{2}e+\sqrt{-64a^3{c}^3{x}^2+48a^2{b}^2{c}^2{x}^2-12a{b}^{4}c{x}^2+b^6{x}^2+32a^{2}b{c}^{2}ex+64a^2{c}^{3}gx-16a{b}^{3}cex-32a{b}^2{c}^{2}gx+2b^{5}ex+4b^{4}cgx+64c_1{a}^2{c}^3-32c_{1}a{b}^2{c}^2+4c_1{b}^{4}c-4a{b}^{2}c{e}^2-16ab{c}^{2}eg-16a{c}^3{g}^2+b^4{e}^2+4b^{3}ceg+4b^2{c}^2{g}^2}}{2c(4ac-b^2)}\end{array}$$

Solved as first order Exact ode

Time used: 0.434 (sec)

To solve an ode of the form

$$\begin{array}{}\text{(A)}& M(x,y)+N(x,y)\frac{dy}{dx}=0\end{array}$$

We assume there exists a function $\varphi (x,y)=c$ where $c$ is constant, that satisfies the ode. Taking derivative of $\varphi$ w.r.t. $x$ gives

$$\frac{d}{dx}\varphi (x,y)=0$$

Hence

$$\begin{array}{}\text{(B)}& \frac{\partial \varphi }{\partial x}+\frac{\partial \varphi }{\partial y}\frac{dy}{dx}=0\end{array}$$

Comparing (A,B) shows that

$$\begin{array}{rl}\frac{\partial \varphi }{\partial x}& =M\\ \frac{\partial \varphi }{\partial y}& =N\end{array}$$

But since $\frac{{\partial }^2\varphi }{\partial x\partial y}=\frac{{\partial }^2\varphi }{\partial y\partial x}$ then for the above to be valid, we require that

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

If the above condition is satisfied, then the original ode is called exact. We still need to determine $\varphi (x,y)$ but at least we know now that we can do that since the condition $\frac{{\partial }^2\varphi }{\partial x\partial y}=\frac{{\partial }^2\varphi }{\partial y\partial x}$ is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

$$\begin{array}{}\text{(1A)}& M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y=0\end{array}$$

Therefore

$$\begin{array}{rl}(bx+2cy+e)\mathrm{d}y& =(-2ax-by+g)\mathrm{d}x\\ \text{(2A)}& (2ax+by-g)\mathrm{d}x+(bx+2cy+e)\mathrm{d}y& =0\end{array}$$

Comparing (1A) and (2A) shows that

$$\begin{array}{rl}M(x,y)& =2ax+by-g\\ N(x,y)& =bx+2cy+e\end{array}$$

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

Using result found above gives

$$\begin{array}{rl}\frac{\partial M}{\partial y}& =\frac{\partial }{\partial y}(2ax+by-g)\\ & =b\end{array}$$

And

$$\begin{array}{rl}\frac{\partial N}{\partial x}& =\frac{\partial }{\partial x}(bx+2cy+e)\\ & =b\end{array}$$

Since $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$, then the ODE is exact The following equations are now set up to solve for the function $\varphi (x,y)$

$$\begin{array}{}\text{(1)}& \frac{\partial \varphi }{\partial x}& =M\text{(2)}& \frac{\partial \varphi }{\partial y}& =N\end{array}$$

Integrating (1) w.r.t. $x$ gives

$$\begin{array}{rl}\int \frac{\partial \varphi }{\partial x}\mathrm{d}x& =\int M\mathrm{d}x\\ \int \frac{\partial \varphi }{\partial x}\mathrm{d}x& =\int 2ax+by-g\mathrm{d}x\\ \text{(3)}& \varphi & =x(ax+by-g)+f(y)\end{array}$$

Where $f(y)$ is used for the constant of integration since $\varphi$ is a function of both $x$ and $y$. Taking derivative of equation (3) w.r.t $y$ gives

$$\begin{array}{}\text{(4)}& \frac{\partial \varphi }{\partial y}=bx+f^{\prime }(y)\end{array}$$

But equation (2) says that $\frac{\partial \varphi }{\partial y}=bx+2cy+e$. Therefore equation (4) becomes

$$\begin{array}{}\text{(5)}& bx+2cy+e=bx+f^{\prime }(y)\end{array}$$

Solving equation (5) for $f^{\prime }(y)$ gives

$$f^{\prime }(y)=2cy+e$$

Integrating the above w.r.t $y$ gives

$$\begin{array}{rl}\int f^{\prime }(y)\mathrm{d}y& =\int (2cy+e)\mathrm{d}y\\ f(y)& =y^{2}c+ey+c_1\end{array}$$

Where $c_1$ is constant of integration. Substituting result found above for $f(y)$ into equation (3) gives $\varphi$

$$\varphi =x(ax+by-g)+y^{2}c+ey+c_1$$

But since $\varphi$ itself is a constant function, then let $\varphi =c_2$ where $c_2$ is new constant and combining $c_1$ and $c_2$ constants into the constant $c_1$ gives the solution as

$$c_1=x(ax+by-g)+y^{2}c+ey$$

Solving for $y$ gives

$$\begin{array}{rl}y& =\frac{-bx-e+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4c_{1}c+e^2}}{2c}\\ y& =-\frac{bx+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4c_{1}c+e^2}+e}{2c}\end{array}$$

Summary of solutions found

$$\begin{array}{rl}y& =\frac{-bx-e+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4c_{1}c+e^2}}{2c}\\ y& =-\frac{bx+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4c_{1}c+e^2}+e}{2c}\end{array}$$

Solved using Lie symmetry for first order ode

Time used: 0.820 (sec)

Writing the ode as

$$\begin{array}{rl}{y}^{\prime }& =-\frac{2ax+by-g}{bx+2cy+e}\\ y^{\prime }& =\omega (x,y)\end{array}$$

The condition of Lie symmetry is the linearized PDE given by

$$\begin{array}{}\text{(A)}& {\eta }_x+\omega ({\eta }_y-{\xi }_x)-{\omega }^2{\xi }_y-{\omega }_x\xi -{\omega }_y\eta =0\end{array}$$

To determine $\xi ,\eta$ then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives

$$\begin{array}{}\text{(1E)}& \xi & =x{a}_2+y{a}_3+a_1\text{(2E)}& \eta & =x{b}_2+y{b}_3+b_1\end{array}$$

Where the unknown coefficients are

$$\{a_1,a_2,a_3,b_1,b_2,b_3\}$$

Substituting equations (1E,2E) and $\omega$ into (A) gives

$$\begin{array}{}\text{(5E)}& b_2-\frac{(2ax+by-g)(b_3-a_2)}{bx+2cy+e}-\frac{{(2ax+by-g)}^2{a}_3}{{(bx+2cy+e)}^2}-(-\frac{2a}{bx+2cy+e}+\frac{(2ax+by-g)b}{{(bx+2cy+e)}^2})(x{a}_2+y{a}_3+a_1)-(-\frac{b}{bx+2cy+e}+\frac{2(2ax+by-g)c}{{(bx+2cy+e)}^2})(x{b}_2+y{b}_3+b_1)=0\end{array}$$

Putting the above in normal form gives

$$-\frac{4a^2{x}^2{a}_3-2ab{x}^2{a}_2+2ab{x}^2{b}_3+4abxy{a}_3+4ac{x}^2{b}_2-8acxy{a}_2+8acxy{b}_3-4ac{y}^2{a}_3-2b^2{x}^2{b}_2+2b^2{y}^2{a}_3-4bcxy{b}_2-2bc{y}^2{a}_2+2bc{y}^2{b}_3-4c^2{y}^2{b}_2+4acx{b}_1-4acy{a}_1-4aex{a}_2+2aex{b}_3-2aey{a}_3-4agx{a}_3-b^{2}x{b}_1+b^{2}y{a}_1-3bex{b}_2-bey{a}_2-bgx{b}_3-3bgy{a}_3-4cey{b}_2-2cgx{b}_2+2cgy{a}_2-4cgy{b}_3-2ae{a}_1-be{b}_1-bg{a}_1-2cg{b}_1-e^2{b}_2+eg{a}_2-eg{b}_3+g^2{a}_3}{{(bx+2cy+e)}^2}=0$$

Setting the numerator to zero gives

$$\begin{array}{}\text{(6E)}& -4a^2{x}^2{a}_3+2ab{x}^2{a}_2-2ab{x}^2{b}_3-4abxy{a}_3-4ac{x}^2{b}_2+8acxy{a}_2-8acxy{b}_3+4ac{y}^2{a}_3+2b^2{x}^2{b}_2-2b^2{y}^2{a}_3+4bcxy{b}_2+2bc{y}^2{a}_2-2bc{y}^2{b}_3+4c^2{y}^2{b}_2-4acx{b}_1+4acy{a}_1+4aex{a}_2-2aex{b}_3+2aey{a}_3+4agx{a}_3+b^{2}x{b}_1-b^{2}y{a}_1+3bex{b}_2+bey{a}_2+bgx{b}_3+3bgy{a}_3+4cey{b}_2+2cgx{b}_2-2cgy{a}_2+4cgy{b}_3+2ae{a}_1+be{b}_1+bg{a}_1+2cg{b}_1+e^2{b}_2-eg{a}_2+eg{b}_3-g^2{a}_3=0\end{array}$$

Looking at the above PDE shows the following are all the terms with $\{x,y\}$ in them.

$$\{x,y\}$$

The following substitution is now made to be able to collect on all terms with $\{x,y\}$ in them

$$\{x=v_1,y=v_2\}$$

The above PDE (6E) now becomes

$$\begin{array}{}\text{(7E)}& -4a^2{a}_3{v}_1^2+2ab{a}_2{v}_1^2-4ab{a}_3{v}_1{v}_2-2ab{b}_3{v}_1^2+8ac{a}_2{v}_1{v}_2+4ac{a}_3{v}_2^2-4ac{b}_2{v}_1^2-8ac{b}_3{v}_1{v}_2-2b^2{a}_3{v}_2^2+2b^2{b}_2{v}_1^2+2bc{a}_2{v}_2^2+4bc{b}_2{v}_1{v}_2-2bc{b}_3{v}_2^2+4c^2{b}_2{v}_2^2+4ac{a}_1{v}_2-4ac{b}_1{v}_1+4ae{a}_2{v}_1+2ae{a}_3{v}_2-2ae{b}_3{v}_1+4ag{a}_3{v}_1-b^2{a}_1{v}_2+b^2{b}_1{v}_1+be{a}_2{v}_2+3be{b}_2{v}_1+3bg{a}_3{v}_2+bg{b}_3{v}_1+4ce{b}_2{v}_2-2cg{a}_2{v}_2+2cg{b}_2{v}_1+4cg{b}_3{v}_2+2ae{a}_1+be{b}_1+bg{a}_1+2cg{b}_1+e^2{b}_2-eg{a}_2+eg{b}_3-g^2{a}_3=0\end{array}$$

Collecting the above on the terms $v_i$ introduced, and these are

$$\{v_1,v_2\}$$

Equation (7E) now becomes

$$\begin{array}{}\text{(8E)}& (-4a^2{a}_3+2ab{a}_2-2ab{b}_3-4ac{b}_2+2b^2{b}_2)v_1^2+(-4ab{a}_3+8ac{a}_2-8ac{b}_3+4bc{b}_2)v_1{v}_2+(-4ac{b}_1+4ae{a}_2-2ae{b}_3+4ag{a}_3+b^2{b}_1+3be{b}_2+bg{b}_3+2cg{b}_2)v_1+(4ac{a}_3-2b^2{a}_3+2bc{a}_2-2bc{b}_3+4c^2{b}_2)v_2^2+(4ac{a}_1+2ae{a}_3-b^2{a}_1+be{a}_2+3bg{a}_3+4ce{b}_2-2cg{a}_2+4cg{b}_3)v_2+2ae{a}_1+be{b}_1+bg{a}_1+2cg{b}_1+e^2{b}_2-eg{a}_2+eg{b}_3-g^2{a}_3=0\end{array}$$

Setting each coefficients in (8E) to zero gives the following equations to solve

$$\begin{array}{rl}-4ab{a}_3+8ac{a}_2-8ac{b}_3+4bc{b}_2& =0\\ 4ac{a}_3-2b^2{a}_3+2bc{a}_2-2bc{b}_3+4c^2{b}_2& =0\\ -4a^2{a}_3+2ab{a}_2-2ab{b}_3-4ac{b}_2+2b^2{b}_2& =0\\ 4ac{a}_1+2ae{a}_3-b^2{a}_1+be{a}_2+3bg{a}_3+4ce{b}_2-2cg{a}_2+4cg{b}_3& =0\\ -4ac{b}_1+4ae{a}_2-2ae{b}_3+4ag{a}_3+b^2{b}_1+3be{b}_2+bg{b}_3+2cg{b}_2& =0\\ 2ae{a}_1+be{b}_1+bg{a}_1+2cg{b}_1+e^2{b}_2-eg{a}_2+eg{b}_3-g^2{a}_3& =0\end{array}$$

Solving the above equations for the unknowns gives

$$\begin{array}{rl}{a}_1& =\frac{2ace{a}_3-b^{2}e{a}_3-bce{b}_3-bcg{a}_3-2c^{2}g{b}_3}{c(4ac-b^2)}\\ a_2& =\frac{b{a}_3+c{b}_3}{c}\\ a_3& =a_3\\ b_1& =\frac{abe{a}_3+2ace{b}_3+2acg{a}_3+bcg{b}_3}{c(4ac-b^2)}\\ b_2& =-\frac{a{a}_3}{c}\\ b_3& =b_3\end{array}$$

Substituting the above solution in the anstaz (1E,2E) (using $1$ as arbitrary value for any unknown in the RHS) gives

$$\begin{array}{rl}\xi & =\frac{4xac-b^{2}x-be-2cg}{4ac-b^2}\\ \eta & =\frac{4acy-b^{2}y+2ae+bg}{4ac-b^2}\end{array}$$

Shifting is now applied to make $\xi =0$ in order to simplify the rest of the computation

$$\begin{array}{rl}\eta & =\eta -\omega (x,y)\xi \\ & =\frac{4acy-b^{2}y+2ae+bg}{4ac-b^2}-(-\frac{2ax+by-g}{bx+2cy+e})\left(\frac{4xac-b^{2}x-be-2cg}{4ac-b^2}\right)\\ & =\frac{8a^{2}c{x}^2-2a{b}^2{x}^2+8abcxy+8a{c}^2{y}^2-2b^{3}xy-2b^{2}c{y}^2+8acey-8acgx-2b^{2}ey+2b^{2}gx+2a{e}^2+2beg+2c{g}^2}{4abcx+8a{c}^{2}y-b^{3}x-2b^{2}cy+4ace-b^{2}e}\\ \xi & =0\end{array}$$

The next step is to determine the canonical coordinates $R,S$. The canonical coordinates map $(x,y)\to (R,S)$ where $(R,S)$ are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.

The characteristic pde which is used to find the canonical coordinates is

$$\begin{array}{}\text{(1)}& \frac{dx}{\xi }& =\frac{dy}{\eta }=dS\end{array}$$

The above comes from the requirements that $(\xi \frac{\partial }{\partial x}+\eta \frac{\partial }{\partial y})S(x,y)=1$. Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable $R$ in the canonical coordinates, where $S(R)$. Since $\xi =0$ then in this special case

$$\begin{array}{r}R=x\end{array}$$

$S$ is found from

$$\begin{array}{rl}S& =\int \frac{1}{\eta }dy\\ & =\int \frac{1}{\frac{8a^{2}c{x}^2-2a{b}^2{x}^2+8abcxy+8a{c}^2{y}^2-2b^{3}xy-2b^{2}c{y}^2+8acey-8acgx-2b^{2}ey+2b^{2}gx+2a{e}^2+2beg+2c{g}^2}{4abcx+8a{c}^{2}y-b^{3}x-2b^{2}cy+4ace-b^{2}e}}dy\end{array}$$

Which results in

$$\begin{array}{rl}S& =\frac{\ln (4a^{2}c{x}^2-a{b}^2{x}^2+4abcxy+4a{c}^2{y}^2-b^{3}xy-b^{2}c{y}^2+4acey-4acgx-b^{2}ey+b^{2}gx+a{e}^2+beg+c{g}^2)}{2}\end{array}$$

Now that $R,S$ are found, we need to setup the ode in these coordinates. This is done by evaluating

$$\begin{array}{}\text{(2)}& \frac{dS}{dR}& =\frac{S_x+\omega (x,y)S_y}{R_x+\omega (x,y)R_y}\end{array}$$

Where in the above $R_x,R_y,S_x,S_y$ are all partial derivatives and $\omega (x,y)$ is the right hand side of the original ode given by

$$\begin{array}{rl}\omega (x,y)& =-\frac{2ax+by-g}{bx+2cy+e}\end{array}$$

Evaluating all the partial derivatives gives

$$\begin{array}{rl}{R}_x& =1\\ R_y& =0\\ S_x& =\frac{(2ax+by-g)(4ac-b^2)}{8a^{2}c{x}^2+(-2b^2{x}^2+8bcxy+8c^2{y}^2+(8ey-8gx)c+2e^2)a-2(by-g)(b^{2}x+(cy+e)b+cg)}\\ S_y& =\frac{(bx+2cy+e)(4ac-b^2)}{8a^{2}c{x}^2+(-2b^2{x}^2+8bcxy+8c^2{y}^2+(8ey-8gx)c+2e^2)a-2(by-g)(b^{2}x+(cy+e)b+cg)}\end{array}$$

Substituting all the above in (2) and simplifying gives the ode in canonical coordinates.

$$\begin{array}{}\text{(2A)}& \frac{dS}{dR}& =0\end{array}$$

We now need to express the RHS as function of $R$ only. This is done by solving for $x,y$ in terms of $R,S$ from the result obtained earlier and simplifying. This gives

$$\begin{array}{rl}\frac{dS}{dR}& =0\end{array}$$

The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates $R,S$.

Since the ode has the form $\frac{d}{dR}S\left(R\right)=f(R)$, then we only need to integrate $f(R)$.

$$\begin{array}{rl}\int dS& =\int 0dR+c_2\\ S\left(R\right)& =c_2\end{array}$$

To complete the solution, we just need to transform the above back to $x,y$ coordinates. This results in

$$\begin{array}{r}\frac{\ln (4a^{2}c{x}^2+(-b^2{x}^2+4bcxy-4cgx+4{(cy+\frac{e}{2})}^2)a-(by-g)(b^{2}x+(cy+e)b+cg))}{2}=c_2\end{array}$$

Summary of solutions found

$$\begin{array}{rl}\frac{\ln (4a^{2}c{x}^2+(-b^2{x}^2+4bcxy-4cgx+4{(cy+\frac{e}{2})}^2)a-(by-g)(b^{2}x+(cy+e)b+cg))}{2}& =c_2\end{array}$$

Solved as first order ode of type dAlembert

Time used: 0.629 (sec)

Let $p=y^{\prime }$ the ode becomes

$$\begin{array}{r}p=-\frac{2ax+by-g}{bx+2cy+e}\end{array}$$

Solving for $y$ from the above results in

$$\begin{array}{}\text{(1)}& y& =-\frac{(bp+2a)x}{2cp+b}-\frac{ep-g}{2cp+b}\end{array}$$

This has the form

$$\begin{array}{}\text{(*)}& y=xf(p)+g(p)\end{array}$$

Where $f,g$ are functions of $p=y^{\prime }(x)$. The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. $x$ gives

$$\begin{array}{rl}p& =f+(x{f}^{\prime }+g^{\prime })\frac{dp}{dx}\\ \text{(2)}& p-f& =(x{f}^{\prime }+g^{\prime })\frac{dp}{dx}\end{array}$$

Comparing the form $y=xf+g$ to (1A) shows that

$$\begin{array}{rl}f& =\frac{-bp-2a}{2cp+b}\\ g& =\frac{-ep+g}{2cp+b}\end{array}$$

Hence (2) becomes

$$\begin{array}{}\text{(2A)}& p-\frac{-bp-2a}{2cp+b}=(-\frac{xb}{2cp+b}+\frac{2xcbp}{{(2cp+b)}^2}+\frac{4xca}{{(2cp+b)}^2}-\frac{e}{2cp+b}+\frac{2cep}{{(2cp+b)}^2}-\frac{2cg}{{(2cp+b)}^2})p^{\prime }\left(x\right)\end{array}$$

The singular solution is found by setting $\frac{dp}{dx}=0$ in the above which gives

$$\begin{array}{r}p-\frac{-bp-2a}{2cp+b}=0\end{array}$$

Solving the above for $p$ results in

$$\begin{array}{rl}{p}_1& =\frac{-b+\sqrt{-4ac+b^2}}{2c}\\ p_2& =-\frac{b+\sqrt{-4ac+b^2}}{2c}\end{array}$$

Substituting these in (1A) and keeping singular solution that verifies the ode gives

$$\begin{array}{r}y=\frac{-bx\sqrt{-4ac+b^2}-4xac+b^{2}x-e\sqrt{-4ac+b^2}+be+2cg}{2c\sqrt{-4ac+b^2}}\\ y=\frac{-bx\sqrt{-4ac+b^2}+4xac-b^{2}x-e\sqrt{-4ac+b^2}-be-2cg}{2\sqrt{-4ac+b^2}c}\end{array}$$

The general solution is found when $\frac{\mathrm{d}p}{\mathrm{d}x}\ne 0$. From eq. (2A). This results in

$$\begin{array}{}\text{(3)}& p^{\prime }\left(x\right)=\frac{p\left(x\right)-\frac{-bp\left(x\right)-2a}{2cp\left(x\right)+b}}{-\frac{xb}{2cp\left(x\right)+b}+\frac{2xcbp\left(x\right)}{{(2cp\left(x\right)+b)}^2}+\frac{4xca}{{(2cp\left(x\right)+b)}^2}-\frac{e}{2cp\left(x\right)+b}+\frac{2cep\left(x\right)}{{(2cp\left(x\right)+b)}^2}-\frac{2cg}{{(2cp\left(x\right)+b)}^2}}\end{array}$$

This ODE is now solved for $p\left(x\right)$. No inversion is needed.

The ode

$$\begin{array}{}\text{(3)}& p^{\prime }\left(x\right)=\frac{2(2cp\left(x\right)+b)(p{\left(x\right)}^{2}c+bp\left(x\right)+a)}{4xac-b^{2}x-be-2cg}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{p}^{\prime }\left(x\right)& =\frac{2(2cp\left(x\right)+b)(p{\left(x\right)}^{2}c+bp\left(x\right)+a)}{4xac-b^{2}x-be-2cg}\\ & =f(x)g(p)\end{array}$$

Where

$$\begin{array}{rl}f(x)& =\frac{2}{4xac-b^{2}x-be-2cg}\\ g(p)& =(2cp+b)(c{p}^2+bp+a)\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(p)}dp& =\int f(x)dx\\ \int \frac{1}{(2cp+b)(c{p}^2+bp+a)}dp& =\int \frac{2}{4xac-b^{2}x-be-2cg}dx\end{array}$$

$$\frac{\ln \left(\frac{{(2cp\left(x\right)+b)}^2}{p{\left(x\right)}^{2}c+bp\left(x\right)+a}\right)}{4ac-b^2}=\frac{2\ln ((4ac-b^2)x-be-2cg)}{4ac-b^2}+c_1$$

Taking the exponential of both sides the solution becomes

$${\left(\frac{{(2cp\left(x\right)+b)}^2}{p{\left(x\right)}^{2}c+bp\left(x\right)+a}\right)}^{\frac{1}{4ac-b^2}}=c_1{((4ac-b^2)x-be-2cg)}^{\frac{2}{4ac-b^2}}$$

Substituing the above solution for $p$ in (2A) gives

$$\begin{array}{rl}y& =\frac{x(-b\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})-2a)}{2c\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+b}+\frac{-e\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+g}{2c\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+b}\end{array}$$

Summary of solutions found

$$\begin{array}{rl}y& =\frac{-bx\sqrt{-4ac+b^2}-4xac+b^{2}x-e\sqrt{-4ac+b^2}+be+2cg}{2c\sqrt{-4ac+b^2}}\\ y& =\frac{-bx\sqrt{-4ac+b^2}+4xac-b^{2}x-e\sqrt{-4ac+b^2}-be-2cg}{2\sqrt{-4ac+b^2}c}\\ y& =\frac{x(-b\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})-2a)}{2c\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+b}+\frac{-e\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+g}{2c\mathrm{RootOf}(-{\left(\frac{{(2\mathit{\text{\_Z}}c+b)}^2}{c{\mathit{\text{\_Z}}}^2+b\mathit{\text{\_Z}}+a}\right)}^{\frac{1}{4ac-b^2}}+c_1{(4xac-b^{2}x-be-2cg)}^{\frac{2}{4ac-b^2}})+b}\end{array}$$

✓ Maple. Time used: 0.105 (sec). Leaf size: 90

ode:=2*a*x+b*y(x)+(2*c*y(x)+b*x+e)*diff(y(x),x) = g; 
dsolve(ode,y(x), singsol=all);
 

$$y\left(x\right)=\frac{-\sqrt{-64(ac-\frac{b^2}{4}){((ax-\frac{g}{2})c-\frac{b(bx+e)}{4})}^2{c}_1^2+4c}+(-4abcx+b^{3}x-4ace+b^{2}e)c_1}{8(ac-\frac{b^2}{4})c{c}_1}$$

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying homogeneous C 
trying homogeneous types: 
trying homogeneous D 
<- homogeneous successful 
<- homogeneous successful`
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & 2ax+by+(2cy+bx+e)y^{\prime }=g\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & y^{\prime }\\ ◻& & \text{Check if ODE is exact}\\ & \circ & \text{ODE is exact if the lhs is the total derivative of a}{C}^2\text{function}\\ & & F^{\prime }(x,y)=0\\ & \circ & \text{Compute derivative of lhs}\\ & & F^{\prime }(x,y)+\left(\frac{\partial }{\partial y}F(x,y)\right)y^{\prime }=0\\ & \circ & \text{Evaluate derivatives}\\ & & b=b\\ & \circ & \text{Condition met, ODE is exact}\\ \bullet & & \text{Exact ODE implies solution will be of this form}\\ & & [F(x,y)=\mathit{C}\mathit{1},M(x,y)=F^{\prime }(x,y),N(x,y)=\frac{\partial }{\partial y}F(x,y)]\\ \bullet & & \text{Solve for}F(x,y)\text{by integrating}M(x,y)\text{with respect to}x\\ & & F(x,y)=\int (2ax+by-g)dx+\mathit{\text{\_F1}}\left(y\right)\\ \bullet & & \text{Evaluate integral}\\ & & F(x,y)=a{x}^2+bxy-gx+\mathit{\text{\_F1}}\left(y\right)\\ \bullet & & \text{Take derivative of}F(x,y)\text{with respect to}y\\ & & N(x,y)=\frac{\partial }{\partial y}F(x,y)\\ \bullet & & \text{Compute derivative}\\ & & bx+2cy+e=bx+\frac{d}{dy}\mathit{\text{\_F1}}\left(y\right)\\ \bullet & & \text{Isolate for}\frac{d}{dy}\mathit{\text{\_F1}}\left(y\right)\\ & & \frac{d}{dy}\mathit{\text{\_F1}}\left(y\right)=2cy+e\\ \bullet & & \text{Solve for}\mathit{\text{\_F1}}\left(y\right)\\ & & \mathit{\text{\_F1}}\left(y\right)=y^{2}c+ey\\ \bullet & & \text{Substitute}\mathit{\text{\_F1}}\left(y\right)\text{into equation for}F(x,y)\\ & & F(x,y)=a{x}^2+bxy+y^{2}c+ey-gx\\ \bullet & & \text{Substitute}F(x,y)\text{into the solution of the ODE}\\ & & a{x}^2+bxy+y^{2}c+ey-gx=\mathit{C}\mathit{1}\\ \bullet & & \text{Solve for}y\\ & & \{y=\frac{-bx-e+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4\mathit{C}\mathit{1}c+e^2}}{2c},y=-\frac{bx+\sqrt{-4ac{x}^2+b^2{x}^2+2bex+4cgx+4\mathit{C}\mathit{1}c+e^2}+e}{2c}\}\end{array}$$

✓ Mathematica. Time used: 17.046 (sec). Leaf size: 132

ode=(2*a*x+b*y[x])+(2*c*y[x]+b*x+e)*D[y[x],x]==g; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$\begin{array}{r}y(x)\to -\frac{\frac{\sqrt{\frac{4cx(g-ax)+b^2{x}^2+2bex+4c^2{c}_1+e^2}{c}}}{\sqrt{\frac{1}{c}}}+bx+e}{2c}\\ y(x)\to -\frac{-\frac{\sqrt{\frac{4cx(g-ax)+b^2{x}^2+2bex+4c^2{c}_1+e^2}{c}}}{\sqrt{\frac{1}{c}}}+bx+e}{2c}\end{array}$$

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
e = symbols("e") 
g = symbols("g") 
y = Function("y") 
ode = Eq(2*a*x + b*y(x) - g + (b*x + 2*c*y(x) + e)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-2*a*x - b*y(x) + g)/(b*x + 2*c*y(x) + e) + Derivative(y(x), x) cannot be solved by the factorable group method