42.1.13 problem Example 3.14

Internal problem ID [8800]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page 114
Problem number : Example 3.14
Date solved : Tuesday, September 30, 2025 at 05:51:43 PM
CAS classification : [_dAlembert]

\begin{align*} x +y y^{\prime }&=a {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 246

ode:=x+y(x)*diff(y(x),x) = a*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {-\frac {\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right ) \operatorname {arcsinh}\left (\frac {y+\sqrt {4 a x +y^{2}}}{2 a}\right )}{2}+x \sqrt {\frac {y \sqrt {4 a x +y^{2}}+2 a^{2}+2 a x +y^{2}}{a^{2}}}+c_1 y+c_1 \sqrt {4 a x +y^{2}}}{\sqrt {\frac {y \sqrt {4 a x +y^{2}}+y^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}-\left (y-\sqrt {4 a x +y^{2}}\right ) \left (c_1 -\operatorname {arcsinh}\left (\frac {-y+\sqrt {4 a x +y^{2}}}{2 a}\right )\right )}{\sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 117

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

\[ \text {Solve}\left [\left \{x=a \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ) \int \frac {\exp \left (-\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right )}{K[1]+\frac {1}{K[1]}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ),y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

Timed Out