77.1.59 problem 78 (page 120)

Internal problem ID [17878]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 78 (page 120)
Date solved : Monday, March 31, 2025 at 04:44:10 PM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.107 (sec). Leaf size: 391

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

\begin{align*} -\frac {c_1 \left (-y+\sqrt {4 a x +y^{2}}\right )}{\sqrt {\frac {-y+\sqrt {4 a x +y^{2}}-2 a}{a}}\, \sqrt {\frac {-y+\sqrt {4 a x +y^{2}}+2 a}{a}}}+x +\frac {\left (-y+\sqrt {4 a x +y^{2}}\right ) \left (-\ln \left (2\right )+\ln \left (\frac {\sqrt {2}\, \sqrt {\frac {y^{2}-y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x}{a^{2}}}\, a +\sqrt {4 a x +y^{2}}-y}{a}\right )\right ) \sqrt {2}}{2 \sqrt {\frac {y^{2}-y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x}{a^{2}}}} &= 0 \\ \frac {c_1 \left (y+\sqrt {4 a x +y^{2}}\right )}{2 \sqrt {\frac {-y-\sqrt {4 a x +y^{2}}-2 a}{a}}\, \sqrt {\frac {-y-\sqrt {4 a x +y^{2}}+2 a}{a}}}+x -\frac {\sqrt {2}\, \left (-\frac {3 \ln \left (2\right )}{2}+\ln \left (\frac {2 \sqrt {\frac {y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x +y^{2}}{a^{2}}}\, a -\left (y+\sqrt {4 a x +y^{2}}\right ) \sqrt {2}}{a}\right )\right ) \left (y+\sqrt {4 a x +y^{2}}\right )}{2 \sqrt {\frac {y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x +y^{2}}{a^{2}}}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.754 (sec). Leaf size: 61

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

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

✗ 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