Problem 6

2.1.6 Problem 6

Solved as ﬁrst order polynomial type ode
Solved using Lie symmetry for ﬁrst order ode
Solved as ﬁrst order ode of type dAlembert
✓Maple
✓Mathematica
✓Sympy

Internal problem ID [4082]
Book : Diﬀerential equations, Shepley L. Ross, 1964
Section : 2.4, page 55
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 05:24:24 PM
CAS classiﬁcation : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Solve

\begin{aligned} y^{'} & = \frac{- 3 x + y - 1}{- 6 x + 2 y + 3} \end{aligned}

Solved as ﬁrst order polynomial type ode

Time used: 0.187 (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^{'} = \frac{a_{1} x + b_{1} y + c_{1}}{a_{2} x + b_{2} y + c_{3}}

Where $a_{1} = 3, b_{1} = - 1, c_{1} = 1, a_{2} = 6, b_{2} = - 2, c_{2} = - 3$ . 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 $\frac{a_{1}}{b_{1}} = \frac{3}{- 1} = - 3$ and $\frac{a_{2}}{b_{2}} = \frac{6}{- 2} = - 3$ . Hence this is case two, where the lines are parallel. Let $U (x) = 3 x - y$ . Solving for $y$ gives

y = 3 x - U (x)

Taking derivative w.r.t $x$ gives

y^{'} = 3 - U^{'} (x)

Substituting the above into the ODE results in the ODE

U (x) + 1 - (2 U (x) - 3) (3 - U^{'} (x)) = 0

(2 U (x) - 3) U^{'} (x) - 5 U (x) + 10 = 0

U^{'} (x) = \frac{5 U (x) - 10}{2 U (x) - 3}

Which is now solved as separable in $U (x)$ . Integrating gives

\begin{aligned} \int \frac{2 U - 3}{5 U - 10} d U & = d x \\ \frac{2 U}{5} + \frac{\ln (U - 2)}{5} & = x + c_{2} \end{aligned}

Singular solutions are found by solving

\begin{aligned} \frac{5 U - 10}{2 U - 3} & = 0 \end{aligned}

for $U (x)$ . This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{array}{r} U (x) = 2 \end{array}

Each of the above solutions is now converted converted back to $y$ . The solution $\frac{2 U (x)}{5} + \frac{\ln (U (x) - 2)}{5} = x + c_{2}$ is converted to $y$ using $U (x) = 3 x - y$ . Which gives

\frac{6 x}{5} - \frac{2 y}{5} + \frac{\ln (3 x - y - 2)}{5} = x + c_{2}

The solution $U (x) = 2$ is converted to $y$ using $U (x) = 3 x - y$ . Which gives

3 x - y = 2

y = - 2 + 3 x

Solving for $y$ gives

\begin{aligned} y & = - 2 + 3 x \\ y & = - \frac{LambertW (2 e^{5 x - 4 + 5 c_{2}})}{2} + 3 x - 2 \end{aligned}

pict — Figure 2.11: Slope ﬁeld $3 x - y + 1 - (6 x - 2 y - 3) y^{'} = 0$

Summary of solutions found

\begin{aligned} y & = - 2 + 3 x \\ y & = - \frac{LambertW (2 e^{5 x - 4 + 5 c_{2}})}{2} + 3 x - 2 \end{aligned}

Solved using Lie symmetry for ﬁrst order ode

Time used: 0.574 (sec)

Writing the ode as

\begin{aligned} y^{'} & = \frac{- 3 x + y - 1}{- 6 x + 2 y + 3} \\ y^{'} & = ω (x, y) \end{aligned}

The condition of Lie symmetry is the linearized PDE given by

\begin{array}{r} (A) & η_{x} + ω (η_{y} - ξ_{x}) - ω^{2} ξ_{y} - ω_{x} ξ - ω_{y} η = 0 \end{array}

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

\begin{aligned} (1E) & ξ & = x a_{2} + y a_{3} + a_{1} \\ (2E) & η & = x b_{2} + y b_{3} + b_{1} \end{aligned}

Where the unknown coeﬃcients are

{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}}

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

\begin{matrix} (5E) & b_{2} + \frac{(- 3 x + y - 1) (b_{3} - a_{2})}{- 6 x + 2 y + 3} - \frac{{(- 3 x + y - 1)}^{2} a_{3}}{{(- 6 x + 2 y + 3)}^{2}} - (- \frac{3}{- 6 x + 2 y + 3} + \frac{- 18 x + 6 y - 6}{{(- 6 x + 2 y + 3)}^{2}}) (x a_{2} + y a_{3} + a_{1}) - (\frac{1}{- 6 x + 2 y + 3} - \frac{2 (- 3 x + y - 1)}{{(- 6 x + 2 y + 3)}^{2}}) (x b_{2} + y b_{3} + b_{1}) = 0 \end{matrix}

Putting the above in normal form gives

- \frac{18 x^{2} a_{2} + 9 x^{2} a_{3} - 36 x^{2} b_{2} - 18 x^{2} b_{3} - 12 x y a_{2} - 6 x y a_{3} + 24 x y b_{2} + 12 x y b_{3} + 2 y^{2} a_{2} + y^{2} a_{3} - 4 y^{2} b_{2} - 2 y^{2} b_{3} - 18 x a_{2} + 6 x a_{3} + 41 x b_{2} + 3 x b_{3} + y a_{2} - 17 y a_{3} - 12 y b_{2} + 4 y b_{3} - 15 a_{1} - 3 a_{2} + a_{3} + 5 b_{1} - 9 b_{2} + 3 b_{3}}{{(6 x - 2 y - 3)}^{2}} = 0

Setting the numerator to zero gives

\begin{matrix} (6E) & - 18 x^{2} a_{2} - 9 x^{2} a_{3} + 36 x^{2} b_{2} + 18 x^{2} b_{3} + 12 x y a_{2} + 6 x y a_{3} - 24 x y b_{2} - 12 x y b_{3} - 2 y^{2} a_{2} - y^{2} a_{3} + 4 y^{2} b_{2} + 2 y^{2} b_{3} + 18 x a_{2} - 6 x a_{3} - 41 x b_{2} - 3 x b_{3} - y a_{2} + 17 y a_{3} + 12 y b_{2} - 4 y b_{3} + 15 a_{1} + 3 a_{2} - a_{3} - 5 b_{1} + 9 b_{2} - 3 b_{3} = 0 \end{matrix}

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{matrix} (7E) & - 18 a_{2} v_{1}^{2} + 12 a_{2} v_{1} v_{2} - 2 a_{2} v_{2}^{2} - 9 a_{3} v_{1}^{2} + 6 a_{3} v_{1} v_{2} - a_{3} v_{2}^{2} + 36 b_{2} v_{1}^{2} - 24 b_{2} v_{1} v_{2} + 4 b_{2} v_{2}^{2} + 18 b_{3} v_{1}^{2} - 12 b_{3} v_{1} v_{2} + 2 b_{3} v_{2}^{2} + 18 a_{2} v_{1} - a_{2} v_{2} - 6 a_{3} v_{1} + 17 a_{3} v_{2} - 41 b_{2} v_{1} + 12 b_{2} v_{2} - 3 b_{3} v_{1} - 4 b_{3} v_{2} + 15 a_{1} + 3 a_{2} - a_{3} - 5 b_{1} + 9 b_{2} - 3 b_{3} = 0 \end{matrix}

Collecting the above on the terms $v_{i}$ introduced, and these are

{v_{1}, v_{2}}

Equation (7E) now becomes

\begin{matrix} (8E) & (- 18 a_{2} - 9 a_{3} + 36 b_{2} + 18 b_{3}) v_{1}^{2} + (12 a_{2} + 6 a_{3} - 24 b_{2} - 12 b_{3}) v_{1} v_{2} + (18 a_{2} - 6 a_{3} - 41 b_{2} - 3 b_{3}) v_{1} + (- 2 a_{2} - a_{3} + 4 b_{2} + 2 b_{3}) v_{2}^{2} + (- a_{2} + 17 a_{3} + 12 b_{2} - 4 b_{3}) v_{2} + 15 a_{1} + 3 a_{2} - a_{3} - 5 b_{1} + 9 b_{2} - 3 b_{3} = 0 \end{matrix}

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

\begin{aligned} - 18 a_{2} - 9 a_{3} + 36 b_{2} + 18 b_{3} & = 0 \\ - 2 a_{2} - a_{3} + 4 b_{2} + 2 b_{3} & = 0 \\ - a_{2} + 17 a_{3} + 12 b_{2} - 4 b_{3} & = 0 \\ 12 a_{2} + 6 a_{3} - 24 b_{2} - 12 b_{3} & = 0 \\ 18 a_{2} - 6 a_{3} - 41 b_{2} - 3 b_{3} & = 0 \\ 15 a_{1} + 3 a_{2} - a_{3} - 5 b_{1} + 9 b_{2} - 3 b_{3} & = 0 \end{aligned}

Solving the above equations for the unknowns gives

\begin{aligned} a_{1} & = a_{1} \\ a_{2} & = 2 b_{2} \\ a_{3} & = - \frac{2 b_{2}}{3} \\ b_{1} & = 3 a_{1} + \frac{10 b_{2}}{3} \\ b_{2} & = b_{2} \\ b_{3} & = - \frac{b_{2}}{3} \end{aligned}

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

\begin{aligned} ξ & = 1 \\ η & = 3 \end{aligned}

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

\begin{aligned} η & = η - ω (x, y) ξ \\ = 3 - (\frac{- 3 x + y - 1}{- 6 x + 2 y + 3}) (1) \\ = \frac{15 x - 5 y - 10}{6 x - 2 y - 3} \\ ξ & = 0 \end{aligned}

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{aligned} (1) & \frac{d x}{ξ} & = \frac{d y}{η} = d S \end{aligned}

The above comes from the requirements that $(ξ \frac{\partial}{\partial x} + η \frac{\partial}{\partial y}) S (x, y) = 1$ . 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 $ξ = 0$ then in this special case

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

$S$ is found from

\begin{aligned} S & = \int \frac{1}{η} d y \\ = \int \frac{1}{\frac{15 x - 5 y - 10}{6 x - 2 y - 3}} d y \end{aligned}

Which results in

\begin{aligned} S & = \frac{2 y}{5} - \frac{\ln (- 3 x + y + 2)}{5} \end{aligned}

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

\begin{aligned} (2) & \frac{d S}{d R} & = \frac{S_{x} + ω (x, y) S_{y}}{R_{x} + ω (x, y) R_{y}} \end{aligned}

Where in the above $R_{x}, R_{y}, S_{x}, S_{y}$ are all partial derivatives and $ω (x, y)$ is the right hand side of the original ode given by

\begin{aligned} ω (x, y) & = \frac{- 3 x + y - 1}{- 6 x + 2 y + 3} \end{aligned}

Evaluating all the partial derivatives gives

\begin{aligned} R_{x} & = 1 \\ R_{y} & = 0 \\ S_{x} & = - \frac{3}{15 x - 5 y - 10} \\ S_{y} & = \frac{2}{5} + \frac{1}{15 x - 5 y - 10} \end{aligned}

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

\begin{aligned} (2A) & \frac{d S}{d R} & = \frac{1}{5} \end{aligned}

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{aligned} \frac{d S}{d R} & = \frac{1}{5} \end{aligned}

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

\begin{aligned} \int d S & = \int \frac{1}{5} d R \\ S (R) & = \frac{R}{5} + c_{2} \end{aligned}

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

\begin{array}{r} \frac{2 y}{5} - \frac{\ln (- 3 x + y + 2)}{5} = \frac{x}{5} + c_{2} \end{array}

Which gives

\begin{array}{r} y = - \frac{LambertW (- 2 e^{5 x - 4 - 5 c_{2}})}{2} + 3 x - 2 \end{array}

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 $(R, S)$

$\frac{d y}{d x} = \frac{- 3 x + y - 1}{- 6 x + 2 y + 3}$		$\frac{d S}{d R} = \frac{1}{5}$
	$\begin{aligned} R & = x \\ S & = \frac{2 y}{5} - \frac{\ln (- 3 x + y + 2)}{5} \end{aligned}$

Summary of solutions found

\begin{aligned} y & = - \frac{LambertW (- 2 e^{5 x - 4 - 5 c_{2}})}{2} + 3 x - 2 \end{aligned}

Solved as ﬁrst order ode of type dAlembert

Time used: 0.414 (sec)

Let $p = y^{'}$ the ode becomes

\begin{array}{r} p = \frac{- 3 x + y - 1}{- 6 x + 2 y + 3} \end{array}

Solving for $y$ from the above results in

\begin{aligned} (1) & y & = \frac{(6 p - 3) x}{- 1 + 2 p} + \frac{- 3 p - 1}{- 1 + 2 p} \end{aligned}

This has the form

\begin{array}{r} (*) & y = x f (p) + g (p) \end{array}

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

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

\begin{aligned} p & = f + (x f^{'} + g^{'}) \frac{d p}{d x} \\ (2) & p - f & = (x f^{'} + g^{'}) \frac{d p}{d x} \end{aligned}

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

\begin{aligned} f & = 3 \\ g & = \frac{- 3 p - 1}{- 1 + 2 p} \end{aligned}

Hence (2) becomes

\begin{matrix} (2A) & p - 3 = (- \frac{3}{- 1 + 2 p} + \frac{6 p}{{(- 1 + 2 p)}^{2}} + \frac{2}{{(- 1 + 2 p)}^{2}}) p^{'} (x) \end{matrix}

The singular solution is found by setting $\frac{d p}{d x} = 0$ in the above which gives

\begin{array}{r} p - 3 = 0 \end{array}

Solving the above for $p$ results in

\begin{aligned} p_{1} & = 3 \end{aligned}

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

\begin{array}{r} y = - 2 + 3 x \end{array}

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

\begin{matrix} (3) & p^{'} (x) = \frac{p (x) - 3}{- \frac{3}{- 1 + 2 p (x)} + \frac{6 p (x)}{{(- 1 + 2 p (x))}^{2}} + \frac{2}{{(- 1 + 2 p (x))}^{2}}} \end{matrix}

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

Integrating gives

\begin{aligned} \int \frac{5}{(p - 3) {(- 1 + 2 p)}^{2}} d p & = d x \\ \frac{1}{- 1 + 2 p} - \frac{\ln (- 1 + 2 p)}{5} + \frac{\ln (p - 3)}{5} & = x + c_{1} \end{aligned}

Singular solutions are found by solving

\begin{aligned} \frac{(p - 3) {(- 1 + 2 p)}^{2}}{5} & = 0 \end{aligned}

for $p (x)$ . This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

\begin{array}{r} p (x) = 3 \\ p (x) = \frac{1}{2} \end{array}

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

\begin{aligned} y & = 3 x + \frac{- 3 e^{RootOf (2 \ln (2 e^{_Z} + 5) e^{_Z} + 10 c_{1} e^{_Z} - 2_Z e^{_Z} + 10 x e^{_Z} + 5 \ln (2 e^{_Z} + 5) + 25 c_{1} - 5_Z + 25 x - 5)} - 10}{2 e^{RootOf (2 \ln (2 e^{_Z} + 5) e^{_Z} + 10 c_{1} e^{_Z} - 2_Z e^{_Z} + 10 x e^{_Z} + 5 \ln (2 e^{_Z} + 5) + 25 c_{1} - 5_Z + 25 x - 5)} + 5} \\ y & = - 2 + 3 x \end{aligned}

Summary of solutions found

\begin{aligned} y & = - 2 + 3 x \\ y & = 3 x + \frac{- 3 e^{RootOf (2 \ln (2 e^{_Z} + 5) e^{_Z} + 10 c_{1} e^{_Z} - 2_Z e^{_Z} + 10 x e^{_Z} + 5 \ln (2 e^{_Z} + 5) + 25 c_{1} - 5_Z + 25 x - 5)} - 10}{2 e^{RootOf (2 \ln (2 e^{_Z} + 5) e^{_Z} + 10 c_{1} e^{_Z} - 2_Z e^{_Z} + 10 x e^{_Z} + 5 \ln (2 e^{_Z} + 5) + 25 c_{1} - 5_Z + 25 x - 5)} + 5} \end{aligned}

✓ Maple. Time used: 0.028 (sec). Leaf size: 23

ode:=3*x-y(x)+1-(6*x-2*y(x)-3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);

y (x) = - \frac{LambertW (- 2 e^{5 x - 4 - 5 c_{1}})}{2} + 3 x - 2

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 
1st order, trying the canonical coordinates of the invariance group 
   -> Calling odsolve with the ODE`, diff(y(x), x) = 3, y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
<- 1st order, canonical coordinates successful 
<- homogeneous successful`

Maple step by step

\begin{array}{lll} Let’s solve \\ 3 x - y (x) + 1 - (6 x - 2 y (x) - 3) (\frac{d}{d x} y (x)) = 0 \\ ∙ & Highest derivative means the order of the ODE is 1 \\ \frac{d}{d x} y (x) \\ ∙ & Solve for the highest derivative \\ \frac{d}{d x} y (x) = - \frac{- 3 x + y (x) - 1}{6 x - 2 y (x) - 3} \end{array}

✓ Mathematica. Time used: 3.076 (sec). Leaf size: 35

ode=(3*x-y[x]+1)-(6*x-2*y[x]-3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

\begin{array}{r} y (x) \to - \frac{1}{2} W (- e^{5 x - 1 + c_{1}}) + 3 x - 2 \\ y (x) \to 3 x - 2 \end{array}

✓ Sympy. Time used: 0.954 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x - (6*x - 2*y(x) - 3)*Derivative(y(x), x) - y(x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)

y (x) = 3 x - \frac{W (C_{1} e^{5 x - 4})}{2} - 2

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