2.1.9 Problem 9

Internal problem ID [4085]
Book : Differential equations, Shepley L. Ross, 1964
Section : 2.4, page 55
Problem number : 9
Date solved : Friday, February 21, 2025 at 08:39:12 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solve

\begin{align*} 3 x -y-6+\left (x +y+2\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=-2 \end{align*}

Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as

\begin{align*} y^{\prime } &= f(x,y)\\ &= \frac {-3 x +y +6}{x +y +2} \end{align*}

The \(x\) domain of \(f(x,y)\) when \(y=-2\) is

\[ \{x <0\boldsymbol {\lor }0<x\} \]

And the point \(x_0 = 2\) is inside this domain. The \(y\) domain of \(f(x,y)\) when \(x=2\) is

\[ \{y <-4\boldsymbol {\lor }-4<y\} \]

And the point \(y_0 = -2\) is inside this domain. Now we will look at the continuity of

\begin{align*} \frac {\partial f}{\partial y} &= \frac {\partial }{\partial y}\left (\frac {-3 x +y +6}{x +y +2}\right ) \\ &= \frac {1}{x +y +2}-\frac {-3 x +y +6}{\left (x +y +2\right )^{2}} \end{align*}

The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y=-2\) is

\[ \{x <0\boldsymbol {\lor }0<x\} \]

And the point \(x_0 = 2\) is inside this domain. The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=2\) is

\[ \{y <-4\boldsymbol {\lor }-4<y\} \]

And the point \(y_0 = -2\) is inside this domain. Therefore solution exists and is unique.

Solved as first order polynomial type ode

Time used: 1.443 (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 \(a_1=-3, b_1=1, c_1 =6, a_2=1, b_2=1, c_2=2\). 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 \(\frac {a_1}{b_1} \neq \frac {a_2}{b_2}\) and the second case when \(\frac {a_1}{b_1} = \frac {a_2}{b_2}\). From the above we see that \(\frac {a_1}{b_1}\neq \frac {a_2}{b_2}\). Hence this is case one where lines are not parallel. Using the transformation

\begin{align*} X &=x-x_0 \\ Y &=y-y_0 \end{align*}

Where the constants \(x_0,y_0\) are obtained by solving the following two linear algebraic equations

\begin{align*} a_1 x_0 + b_1 y_0 + c_1 &= 0\\ a_2 x_0 + b_2 y_0 + c_2 &= 0 \end{align*}

Substituting the values for \(a_1,b_1,c_1,a_2,b_2,c_2\) gives

\begin{align*} -3 x_{0} +y_{0} +6 &= 0 \\ x_{0} +y_{0} +2 &= 0 \\ \end{align*}

Solving for \(x_0,y_0\) from the above gives

\begin{align*} x_0 &= 1 \\ y_0 &= -3 \end{align*}

Therefore the transformation becomes

\begin{align*} X &=x-1 \\ Y &=y+3 \end{align*}

Using this transformation in \(3 x -y-6+\left (x +y+2\right ) y^{\prime } = 0\) result in

\begin{align*} \frac {dY}{dX} &= \frac {-3 X +Y}{X +Y} \end{align*}

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

\begin{align*} Y' &= F(X,Y)\\ &= \frac {-3 X +Y}{X +Y}\tag {1} \end{align*}

An ode of the form \(Y' = \frac {M(X,Y)}{N(X,Y)}\) 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 \(M=-3 X +Y\) and \(N=X +Y\) 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

\[ \frac { \mathop {\mathrm {d}Y}}{\mathop {\mathrm {d}X}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u \]

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

\begin{align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u &= \frac {u -3}{u +1}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}} &= \frac {\frac {u \left (X \right )-3}{u \left (X \right )+1}-u \left (X \right )}{X} \end{align*}

Or

\[ \frac {d}{d X}u \left (X \right )-\frac {\frac {u \left (X \right )-3}{u \left (X \right )+1}-u \left (X \right )}{X} = 0 \]

Or

\[ \left (\frac {d}{d X}u \left (X \right )\right ) X u \left (X \right )+\left (\frac {d}{d X}u \left (X \right )\right ) X +u \left (X \right )^{2}+3 = 0 \]

Or

\[ X \left (u \left (X \right )+1\right ) \left (\frac {d}{d X}u \left (X \right )\right )+u \left (X \right )^{2}+3 = 0 \]

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

The ode

\begin{equation} \frac {d}{d X}u \left (X \right ) = -\frac {u \left (X \right )^{2}+3}{X \left (u \left (X \right )+1\right )} \end{equation}

is separable as it can be written as

\begin{align*} \frac {d}{d X}u \left (X \right )&= -\frac {u \left (X \right )^{2}+3}{X \left (u \left (X \right )+1\right )}\\ &= f(X) g(u) \end{align*}

Where

\begin{align*} f(X) &= -\frac {1}{X}\\ g(u) &= \frac {u^{2}+3}{u +1} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(X) \,dX} \\ \int { \frac {u +1}{u^{2}+3}\,du} &= \int { -\frac {1}{X} \,dX} \\ \end{align*}

\[ \frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3}=\ln \left (\frac {1}{X}\right )+c_2 \]

We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or

\[ \frac {u^{2}+3}{u +1}=0 \]

for \(u \left (X \right )\) gives

\begin{align*} u \left (X \right )&=-i \sqrt {3}\\ u \left (X \right )&=i \sqrt {3} \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

\begin{align*} \frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3} &= \ln \left (\frac {1}{X}\right )+c_2 \\ u \left (X \right ) &= -i \sqrt {3} \\ u \left (X \right ) &= i \sqrt {3} \\ \end{align*}

Converting \(\frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3} = \ln \left (\frac {1}{X}\right )+c_2\) back to \(Y \left (X \right )\) gives

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {Y \left (X \right )^{2}+3 X^{2}}{X^{2}}\right )+2 \arctan \left (\frac {Y \left (X \right ) \sqrt {3}}{3 X}\right )\right )}{6} = \ln \left (\frac {1}{X}\right )+c_2 \end{align*}

Converting \(u \left (X \right ) = -i \sqrt {3}\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = -i X \sqrt {3} \end{align*}

Converting \(u \left (X \right ) = i \sqrt {3}\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = i X \sqrt {3} \end{align*}

The solution is implicit \(\frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {Y \left (X \right )^{2}+3 X^{2}}{X^{2}}\right )+2 \arctan \left (\frac {Y \left (X \right ) \sqrt {3}}{3 X}\right )\right )}{6} = \ln \left (\frac {1}{X}\right )+c_2\). Replacing \(Y=y-y_0, X=x-x_0\) gives

\[ \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {3 \left (x -1\right )^{2}+\left (y+3\right )^{2}}{\left (x -1\right )^{2}}\right )+2 \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )\right )}{6} = \ln \left (\frac {1}{x -1}\right )+c_2 \]

The solution is

\[ Y \left (X \right ) = -i X \sqrt {3} \]

Replacing \(Y=y-y_0, X=x-x_0\) gives

\[ y+3 = -i \left (x -1\right ) \sqrt {3} \]

Or

\[ y = -i \left (x -1\right ) \sqrt {3}-3 \]

The solution is

\[ Y \left (X \right ) = i X \sqrt {3} \]

Replacing \(Y=y-y_0, X=x-x_0\) gives

\[ y+3 = i \left (x -1\right ) \sqrt {3} \]

Or

\[ y = i \left (x -1\right ) \sqrt {3}-3 \]

Solving for the constant of integration from initial conditions, the solution becomes

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {3 \left (x -1\right )^{2}+\left (y+3\right )^{2}}{\left (x -1\right )^{2}}\right )+2 \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )\right )}{6} = \ln \left (\frac {1}{x -1}\right )+\frac {\sqrt {3}\, \left (6 \sqrt {3}\, \ln \left (2\right )+\pi \right )}{18} \end{align*}

The solution

\[ y = -i \left (x -1\right ) \sqrt {3}-3 \]

was found not to satisfy the ode or the IC. Hence it is removed. The solution

\[ y = i \left (x -1\right ) \sqrt {3}-3 \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 2.17: Slope field \(3 x -y-6+\left (x +y+2\right ) y^{\prime } = 0\)

Summary of solutions found

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {3 \left (x -1\right )^{2}+\left (y+3\right )^{2}}{\left (x -1\right )^{2}}\right )+2 \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )\right )}{6} &= \ln \left (\frac {1}{x -1}\right )+\frac {\sqrt {3}\, \left (6 \sqrt {3}\, \ln \left (2\right )+\pi \right )}{18} \\ \end{align*}

Solved as first order homogeneous class Maple C ode

Time used: 0.972 (sec)

Let \(Y = y -y_{0}\) and \(X = x -x_{0}\) then the above is transformed to new ode in \(Y(X)\)

\[ \frac {d}{d X}Y \left (X \right ) = \frac {-3 X -3 x_{0} +Y \left (X \right )+y_{0} +6}{X +x_{0} +Y \left (X \right )+y_{0} +2} \]

Solving for possible values of \(x_{0}\) and \(y_{0}\) which makes the above ode a homogeneous ode results in

\begin{align*} x_{0}&=1\\ y_{0}&=-3 \end{align*}

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

\begin{align*} \frac {d}{d X}Y \left (X \right ) = \frac {-3 X +Y \left (X \right )}{X +Y \left (X \right )} \end{align*}

In canonical form, the ODE is

\begin{align*} Y' &= F(X,Y)\\ &= \frac {-3 X +Y}{X +Y}\tag {1} \end{align*}

An ode of the form \(Y' = \frac {M(X,Y)}{N(X,Y)}\) 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 \(M=-3 X +Y\) and \(N=X +Y\) 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

\[ \frac { \mathop {\mathrm {d}Y}}{\mathop {\mathrm {d}X}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u \]

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

\begin{align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}}X + u &= \frac {u -3}{u +1}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}X}} &= \frac {\frac {u \left (X \right )-3}{u \left (X \right )+1}-u \left (X \right )}{X} \end{align*}

Or

\[ \frac {d}{d X}u \left (X \right )-\frac {\frac {u \left (X \right )-3}{u \left (X \right )+1}-u \left (X \right )}{X} = 0 \]

Or

\[ \left (\frac {d}{d X}u \left (X \right )\right ) X u \left (X \right )+\left (\frac {d}{d X}u \left (X \right )\right ) X +u \left (X \right )^{2}+3 = 0 \]

Or

\[ X \left (u \left (X \right )+1\right ) \left (\frac {d}{d X}u \left (X \right )\right )+u \left (X \right )^{2}+3 = 0 \]

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

The ode

\begin{equation} \frac {d}{d X}u \left (X \right ) = -\frac {u \left (X \right )^{2}+3}{X \left (u \left (X \right )+1\right )} \end{equation}

is separable as it can be written as

\begin{align*} \frac {d}{d X}u \left (X \right )&= -\frac {u \left (X \right )^{2}+3}{X \left (u \left (X \right )+1\right )}\\ &= f(X) g(u) \end{align*}

Where

\begin{align*} f(X) &= -\frac {1}{X}\\ g(u) &= \frac {u^{2}+3}{u +1} \end{align*}

Integrating gives

\begin{align*} \int { \frac {1}{g(u)} \,du} &= \int { f(X) \,dX} \\ \int { \frac {u +1}{u^{2}+3}\,du} &= \int { -\frac {1}{X} \,dX} \\ \end{align*}

\[ \frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3}=\ln \left (\frac {1}{X}\right )+c_1 \]

We now need to find the singular solutions, these are found by finding for what values \(g(u)\) is zero, since we had to divide by this above. Solving \(g(u)=0\) or

\[ \frac {u^{2}+3}{u +1}=0 \]

for \(u \left (X \right )\) gives

\begin{align*} u \left (X \right )&=-i \sqrt {3}\\ u \left (X \right )&=i \sqrt {3} \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

\begin{align*} \frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3} &= \ln \left (\frac {1}{X}\right )+c_1 \\ u \left (X \right ) &= -i \sqrt {3} \\ u \left (X \right ) &= i \sqrt {3} \\ \end{align*}

Converting \(\frac {\ln \left (u \left (X \right )^{2}+3\right )}{2}+\frac {\sqrt {3}\, \arctan \left (\frac {u \left (X \right ) \sqrt {3}}{3}\right )}{3} = \ln \left (\frac {1}{X}\right )+c_1\) back to \(Y \left (X \right )\) gives

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {Y \left (X \right )^{2}+3 X^{2}}{X^{2}}\right )+2 \arctan \left (\frac {Y \left (X \right ) \sqrt {3}}{3 X}\right )\right )}{6} = \ln \left (\frac {1}{X}\right )+c_1 \end{align*}

Converting \(u \left (X \right ) = -i \sqrt {3}\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = -i X \sqrt {3} \end{align*}

Converting \(u \left (X \right ) = i \sqrt {3}\) back to \(Y \left (X \right )\) gives

\begin{align*} Y \left (X \right ) = i X \sqrt {3} \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {Y \left (X \right )^{2}+3 X^{2}}{X^{2}}\right )+2 \arctan \left (\frac {Y \left (X \right ) \sqrt {3}}{3 X}\right )\right )}{6} = \ln \left (\frac {1}{X}\right )+c_1\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x +x_{0} \end{align*}

Or

\begin{align*} Y &= y -3\\ X &= x +1 \end{align*}

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

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {3 \left (x -1\right )^{2}+\left (y+3\right )^{2}}{\left (x -1\right )^{2}}\right )+2 \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )\right )}{6} = \ln \left (\frac {1}{x -1}\right )+c_1 \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} Y \left (X \right ) = -i X \sqrt {3}\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x +x_{0} \end{align*}

Or

\begin{align*} Y &= y -3\\ X &= x +1 \end{align*}

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

\begin{align*} y+3 = -i \left (x -1\right ) \sqrt {3} \end{align*}

Using the solution for \(Y(X)\)

\begin{align*} Y \left (X \right ) = i X \sqrt {3}\tag {A} \end{align*}

And replacing back terms in the above solution using

\begin{align*} Y &= y +y_{0}\\ X &= x +x_{0} \end{align*}

Or

\begin{align*} Y &= y -3\\ X &= x +1 \end{align*}

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

\begin{align*} y+3 = i \left (x -1\right ) \sqrt {3} \end{align*}

Solving for the constant of integration from initial conditions, the solution becomes

\begin{align*} \frac {\sqrt {3}\, \left (\sqrt {3}\, \ln \left (\frac {3 \left (x -1\right )^{2}+\left (y+3\right )^{2}}{\left (x -1\right )^{2}}\right )+2 \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )\right )}{6} = \ln \left (\frac {1}{x -1}\right )+\frac {\sqrt {3}\, \left (6 \sqrt {3}\, \ln \left (2\right )+\pi \right )}{18} \end{align*}

The solution

\[ y+3 = -i \left (x -1\right ) \sqrt {3} \]

was found not to satisfy the ode or the IC. Hence it is removed. The solution

\[ y+3 = i \left (x -1\right ) \sqrt {3} \]

was found not to satisfy the ode or the IC. Hence it is removed.

Figure 2.18: Slope field \(3 x -y-6+\left (x +y+2\right ) y^{\prime } = 0\)

Solved using Lie symmetry for first order ode

Time used: 1.075 (sec)

Writing the ode as

\begin{align*} y^{\prime }&=\frac {-3 x +y +6}{x +y +2}\\ y^{\prime }&= \omega \left ( x,y\right ) \end{align*}

The condition of Lie symmetry is the linearized PDE given by

\begin{align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end{align*}

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

\begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*}

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{equation} \tag{5E} b_{2}+\frac {\left (-3 x +y +6\right ) \left (b_{3}-a_{2}\right )}{x +y +2}-\frac {\left (-3 x +y +6\right )^{2} a_{3}}{\left (x +y +2\right )^{2}}-\left (-\frac {3}{x +y +2}-\frac {-3 x +y +6}{\left (x +y +2\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (\frac {1}{x +y +2}-\frac {-3 x +y +6}{\left (x +y +2\right )^{2}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation}

Putting the above in normal form gives

\[ \frac {3 x^{2} a_{2}-9 x^{2} a_{3}-3 x^{2} b_{2}-3 x^{2} b_{3}+6 x y a_{2}+6 x y a_{3}+2 x y b_{2}-6 x y b_{3}-y^{2} a_{2}+3 y^{2} a_{3}+y^{2} b_{2}+y^{2} b_{3}+12 x a_{2}+36 x a_{3}-4 x b_{1}+8 x b_{2}+4 y a_{1}-8 y a_{2}+4 y b_{2}+12 y b_{3}+12 a_{1}-12 a_{2}-36 a_{3}+4 b_{1}+4 b_{2}+12 b_{3}}{\left (x +y +2\right )^{2}} = 0 \]

Setting the numerator to zero gives

\begin{equation} \tag{6E} 3 x^{2} a_{2}-9 x^{2} a_{3}-3 x^{2} b_{2}-3 x^{2} b_{3}+6 x y a_{2}+6 x y a_{3}+2 x y b_{2}-6 x y b_{3}-y^{2} a_{2}+3 y^{2} a_{3}+y^{2} b_{2}+y^{2} b_{3}+12 x a_{2}+36 x a_{3}-4 x b_{1}+8 x b_{2}+4 y a_{1}-8 y a_{2}+4 y b_{2}+12 y b_{3}+12 a_{1}-12 a_{2}-36 a_{3}+4 b_{1}+4 b_{2}+12 b_{3} = 0 \end{equation}

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{equation} \tag{7E} 3 a_{2} v_{1}^{2}+6 a_{2} v_{1} v_{2}-a_{2} v_{2}^{2}-9 a_{3} v_{1}^{2}+6 a_{3} v_{1} v_{2}+3 a_{3} v_{2}^{2}-3 b_{2} v_{1}^{2}+2 b_{2} v_{1} v_{2}+b_{2} v_{2}^{2}-3 b_{3} v_{1}^{2}-6 b_{3} v_{1} v_{2}+b_{3} v_{2}^{2}+4 a_{1} v_{2}+12 a_{2} v_{1}-8 a_{2} v_{2}+36 a_{3} v_{1}-4 b_{1} v_{1}+8 b_{2} v_{1}+4 b_{2} v_{2}+12 b_{3} v_{2}+12 a_{1}-12 a_{2}-36 a_{3}+4 b_{1}+4 b_{2}+12 b_{3} = 0 \end{equation}

Collecting the above on the terms \(v_i\) introduced, and these are

\[ \{v_{1}, v_{2}\} \]

Equation (7E) now becomes

\begin{equation} \tag{8E} \left (3 a_{2}-9 a_{3}-3 b_{2}-3 b_{3}\right ) v_{1}^{2}+\left (6 a_{2}+6 a_{3}+2 b_{2}-6 b_{3}\right ) v_{1} v_{2}+\left (12 a_{2}+36 a_{3}-4 b_{1}+8 b_{2}\right ) v_{1}+\left (-a_{2}+3 a_{3}+b_{2}+b_{3}\right ) v_{2}^{2}+\left (4 a_{1}-8 a_{2}+4 b_{2}+12 b_{3}\right ) v_{2}+12 a_{1}-12 a_{2}-36 a_{3}+4 b_{1}+4 b_{2}+12 b_{3} = 0 \end{equation}

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

\begin{align*} 4 a_{1}-8 a_{2}+4 b_{2}+12 b_{3}&=0\\ -a_{2}+3 a_{3}+b_{2}+b_{3}&=0\\ 3 a_{2}-9 a_{3}-3 b_{2}-3 b_{3}&=0\\ 6 a_{2}+6 a_{3}+2 b_{2}-6 b_{3}&=0\\ 12 a_{2}+36 a_{3}-4 b_{1}+8 b_{2}&=0\\ 12 a_{1}-12 a_{2}-36 a_{3}+4 b_{1}+4 b_{2}+12 b_{3}&=0 \end{align*}

Solving the above equations for the unknowns gives

\begin{align*} a_{1}&=-b_{3}+3 a_{3}\\ a_{2}&=b_{3}\\ a_{3}&=a_{3}\\ b_{1}&=3 a_{3}+3 b_{3}\\ b_{2}&=-3 a_{3}\\ b_{3}&=b_{3} \end{align*}

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

\begin{align*} \xi &= x -1 \\ \eta &= y +3 \\ \end{align*}

The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) 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{align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end{align*}

The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). 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 \(S(R)\). Therefore

\begin{align*} \frac {dy}{dx} &= \frac {\eta }{\xi }\\ &= \frac {y +3}{x -1}\\ &= \frac {y +3}{x -1} \end{align*}

This is easily solved to give

\begin{align*} y = -3+\left (x -1\right ) c_1 \end{align*}

Where now the coordinate \(R\) is taken as the constant of integration. Hence

\begin{align*} R &= \frac {y +3}{x -1} \end{align*}

And \(S\) is found from

\begin{align*} dS &= \frac {dx}{\xi } \\ &= \frac {dx}{x -1} \end{align*}

Integrating gives

\begin{align*} S &= \int { \frac {dx}{T}}\\ &= \ln \left (x -1\right ) \end{align*}

Where the constant of integration is set to zero as we just need one solution. Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating

\begin{align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end{align*}

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{align*} \omega (x,y) &= \frac {-3 x +y +6}{x +y +2} \end{align*}

Evaluating all the partial derivatives gives

\begin{align*} R_{x} &= \frac {-y -3}{\left (x -1\right )^{2}}\\ R_{y} &= \frac {1}{x -1}\\ S_{x} &= \frac {1}{x -1}\\ S_{y} &= 0 \end{align*}

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

\begin{align*} \frac {dS}{dR} &= -\frac {\left (x +y +2\right ) \left (x -1\right )}{3 x^{2}+y^{2}-6 x +6 y +12}\tag {2A} \end{align*}

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{align*} \frac {dS}{dR} &= \frac {-R -1}{R^{2}+3} \end{align*}

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 \(\frac {d}{d R}S \left (R \right )=f(R)\), then we only need to integrate \(f(R)\).

\begin{align*} \int {dS} &= \int {-\frac {R +1}{R^{2}+3}\, dR}\\ S \left (R \right ) &= -\frac {\ln \left (R^{2}+3\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {R \sqrt {3}}{3}\right )}{3} + c_2 \end{align*}

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

\begin{align*} \ln \left (x -1\right ) = -\frac {\ln \left (\frac {\left (y+3\right )^{2}}{\left (x -1\right )^{2}}+3\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )}{3}+c_2 \end{align*}

The following diagram shows solution curves of the original ode and how they transform in the canonical coordinates space using the mapping shown.

Original ode in \(x,y\) coordinates	Canonical coordinates transformation	ODE in canonical coordinates \((R,S)\)
\( \frac {dy}{dx} = \frac {-3 x +y +6}{x +y +2}\)		\( \frac {d S}{d R} = \frac {-R -1}{R^{2}+3}\)
	\(\!\begin {aligned} R&= \frac {y +3}{x -1}\\ S&= \ln \left (x -1\right ) \end {aligned} \)

Solving for the constant of integration from initial conditions, the solution becomes

\begin{align*} \ln \left (x -1\right ) = -\frac {\ln \left (\frac {\left (y+3\right )^{2}}{\left (x -1\right )^{2}}+3\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )}{3}+\ln \left (2\right )+\frac {\sqrt {3}\, \pi }{18} \end{align*}

Figure 2.19: Slope field \(3 x -y-6+\left (x +y+2\right ) y^{\prime } = 0\)

Summary of solutions found

\begin{align*} \ln \left (x -1\right ) &= -\frac {\ln \left (\frac {\left (y+3\right )^{2}}{\left (x -1\right )^{2}}+3\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (y+3\right ) \sqrt {3}}{3 x -3}\right )}{3}+\ln \left (2\right )+\frac {\sqrt {3}\, \pi }{18} \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [3 x -y \left (x \right )-6+\left (x +y \left (x \right )+2\right ) \left (\frac {d}{d x}y \left (x \right )\right )=0, y \left (2\right )=-2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-3 x +y \left (x \right )+6}{x +y \left (x \right )+2} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (2\right )=-2 \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \end {array} \]

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 dsolve solution

Solving time : 2.393 (sec)
Leaf size : 51

dsolve([3*x-y(x)-6+(x+y(x)+2)*diff(y(x),x) = 0,op([y(2) = -2])],y(x),singsol=all)
 

\[ y \left (x \right ) = -3-\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-3 \sqrt {3}\, \ln \left (3\right )+6 \sqrt {3}\, \ln \left (2\right )-3 \sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (x -1\right )^{2}\right )+\pi +6 \textit {\_Z} \right )\right ) \left (x -1\right ) \]

✓Mathematica DSolve solution

Solving time : 0.138 (sec)
Leaf size : 90

DSolve[{(3*x-y[x]-6)+(x+y[x]+2)*D[y[x],x]==0,y[2]==-2},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\arctan \left (\frac {-y(x)+3 x-6}{\sqrt {3} (y(x)+x+2)}\right )}{\sqrt {3}}+\log (2)=\frac {1}{2} \log \left (\frac {3 x^2+y(x)^2+6 y(x)-6 x+12}{(x-1)^2}\right )+\log (x-1)+\frac {1}{18} \left (\sqrt {3} \pi +18 \log (2)-9 \log (4)\right ),y(x)\right ] \]

✓Sympy solution

Solving time : 4.887 (sec)
Leaf size : 58

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x + (x + y(x) + 2)*Derivative(y(x), x) - y(x) - 6,0) 
ics = {y(2): -2} 
dsolve(ode,func=y(x),ics=ics)
 

Eq(log(x - 1), -log(sqrt(3 + (y(x) + 3)**2/(x - 1)**2)) - 
sqrt(3)*atan(sqrt(3)*(y(x) + 3)/(3*(x - 1)))/3 + sqrt(3)*pi/18 + 
log(2))
 

\[ \log {\left (x - 1 \right )} = - \log {\left (\sqrt {3 + \frac {\left (y{\left (x \right )} + 3\right )^{2}}{\left (x - 1\right )^{2}}} \right )} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (y{\left (x \right )} + 3\right )}{3 \left (x - 1\right )} \right )}}{3} + \frac {\sqrt {3} \pi }{18} + \log {\left (2 \right )} \]