2.1.3 Problem 3
Internal
problem
ID
[4088]
Book
:
Applied
Differential
equations,
Newby
Curle.
Van
Nostrand
Reinhold.
1972
Section
:
Examples,
page
35
Problem
number
:
3
Date
solved
:
Wednesday, November 26, 2025 at 03:33:46 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
Solved using first_order_ode_dAlembert
Time used: 0.060 (sec)
Solve
\begin{align*}
y-x&={y^{\prime }}^{2} \left (1-\frac {2 y^{\prime }}{3}\right ) \\
\end{align*}
Let
\(p=y^{\prime }\) the ode becomes
\begin{align*} y -x = p^{2} \left (1-\frac {2 p}{3}\right ) \end{align*}
Solving for \(y\) from the above results in
\begin{align*}
\tag{1} y &= p^{2}-\frac {2}{3} p^{3}+x \\
\end{align*}
This has the form
\begin{align*} y=x f(p)+g(p)\tag {*} \end{align*}
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{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}
Comparing the form \(y=x f + g\) to (1A) shows that
\begin{align*} f &= 1\\ g &= p^{2}-\frac {2}{3} p^{3} \end{align*}
Hence (2) becomes
\begin{equation}
\tag{2A} p -1 = \left (-2 p^{2}+2 p \right ) p^{\prime }\left (x \right )
\end{equation}
The singular solution is found by setting
\(\frac {dp}{dx}=0\) in the above which gives
\begin{align*} p -1 = 0 \end{align*}
Solving the above for \(p\) results in
\begin{align*} p_{1} &=1 \end{align*}
Substituting these in (1A) and keeping singular solution that verifies the ode gives
\begin{align*} y = x +\frac {1}{3} \end{align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in
\begin{equation}
\tag{3} p^{\prime }\left (x \right ) = \frac {p \left (x \right )-1}{-2 p \left (x \right )^{2}+2 p \left (x \right )}
\end{equation}
This ODE is now solved for
\(p \left (x \right )\).
No inversion is needed.
Integrating gives
\begin{align*} \int -2 p d p &= dx\\ -p^{2}&= x +c_1 \end{align*}
Substituing the above solution for \(p\) in (2A) gives
\begin{align*}
y &= -c_1 -\frac {2 \left (-x -c_1 \right )^{{3}/{2}}}{3} \\
\end{align*}
Simplifying the above gives
\begin{align*}
y &= x +\frac {1}{3} \\
y &= \frac {\left (2 x +2 c_1 \right ) \sqrt {-x -c_1}}{3}-c_1 \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= x +\frac {1}{3} \\
y &= \frac {\left (2 x +2 c_1 \right ) \sqrt {-x -c_1}}{3}-c_1 \\
\end{align*}
Solved using first_order_ode_parametric method
Time used: 0.105 (sec)
Solve
\begin{align*}
y-x&={y^{\prime }}^{2} \left (1-\frac {2 y^{\prime }}{3}\right ) \\
\end{align*}
Let
\(y^{\prime }\) be a parameter
\(\lambda \). The ode becomes
\begin{align*} y -x -\lambda ^{2} \left (1-\frac {2 \lambda }{3}\right ) = 0 \end{align*}
Isolating \(y\) gives
\begin{align*} y&=x +\lambda ^{2}-\frac {2}{3} \lambda ^{3}\\ &=x +\lambda ^{2}-\frac {2}{3} \lambda ^{3}\\ &=F \left (x , \lambda \right ) \end{align*}
Now we generate an ode in \(x \left (\lambda \right )\) using
\begin{align*} \frac {d}{d \lambda }x \left (\lambda \right ) &= \frac { \frac {\partial F}{\partial \lambda }} { \lambda -\frac {\partial F}{\partial x} } \\ &= \frac {-2 \lambda ^{2}+2 \lambda }{\lambda -1}\\ &= -2 \lambda \end{align*}
Which is now solved for \(x\).
Solve Since the ode has the form \(\frac {d}{d \lambda }x \left (\lambda \right )=f(\lambda )\), then we only need to integrate \(f(\lambda )\).
\begin{align*} \int {dx} &= \int {-2 \lambda \, d\lambda }\\ x \left (\lambda \right ) &= -\lambda ^{2} + c_1 \end{align*}
Now that we found solution \(x\) we have two equations with parameter \(\lambda \). They are
\begin{align*}
y &= x +\lambda ^{2}-\frac {2}{3} \lambda ^{3} \\
x &= -\lambda ^{2}+c_1 \\
\end{align*}
Eliminating
\(\lambda \) gives
the solution for
\(y\). Simplifying the above gives
\begin{align*}
y &= c_1 -\frac {2 \sqrt {\left (-x +c_1 \right )^{3}}}{3} \\
y &= c_1 +\frac {2 \sqrt {\left (-x +c_1 \right )^{3}}}{3} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= c_1 -\frac {2 \sqrt {\left (-x +c_1 \right )^{3}}}{3} \\
y &= c_1 +\frac {2 \sqrt {\left (-x +c_1 \right )^{3}}}{3} \\
\end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 49
ode:=-x+y(x) = diff(y(x),x)^2*(1-2/3*diff(y(x),x));
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= x +\frac {1}{3} \\
y &= \frac {\left (2 x -2 c_1 \right ) \sqrt {c_1 -x}}{3}+c_1 \\
y &= \frac {\left (-2 x +2 c_1 \right ) \sqrt {c_1 -x}}{3}+c_1 \\
\end{align*}
Maple trace
Methods for first order ODEs:
*** Sublevel 2 ***
Methods for first order ODEs:
-> Solving 1st order ODE of high degree, 1st attempt
trying 1st order WeierstrassP solution for high degree ODE
trying 1st order WeierstrassPPrime solution for high degree ODE
trying 1st order JacobiSN solution for high degree ODE
trying 1st order ODE linearizable_by_differentiation
trying differential order: 1; missing variables
trying dAlembert
<- dAlembert successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right )-x =\left (\frac {d}{d x}y \left (x \right )\right )^{2} \left (1-\frac {2 \frac {d}{d x}y \left (x \right )}{3}\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} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2}, \frac {d}{d x}y \left (x \right )=-\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{4}-\frac {1}{4 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}-\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}\right )}{2}, \frac {d}{d x}y \left (x \right )=-\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{4}-\frac {1}{4 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}-\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{4}-\frac {1}{4 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}-\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{4}-\frac {1}{4 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}+\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}{2}-\frac {1}{2 \left (1-6 y \left (x \right )+6 x +2 \sqrt {-3 y \left (x \right )+3 x +9 y \left (x \right )^{2}-18 y \left (x \right ) x +9 x^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
✓ Mathematica. Time used: 160.121 (sec). Leaf size: 14234
ode=y[x]-x==D[y[x],x]^2*(1-2/3* D[y[x],x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x - (1 - 2*Derivative(y(x), x)/3)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out