54.1.394 problem 405

Internal problem ID [11708]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 405
Date solved : Tuesday, September 30, 2025 at 10:11:41 PM
CAS classification : [_dAlembert]

\begin{align*} a {y^{\prime }}^{2}+y y^{\prime }-x&=0 \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 391

ode:=a*diff(y(x),x)^2+y(x)*diff(y(x),x)-x = 0; 
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 {\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 -\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right )}{a}\right )\right ) \left (y+\sqrt {4 a x +y^{2}}\right ) \sqrt {2}}{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: 0.205 (sec). Leaf size: 129

ode=-x + y[x]*D[y[x],x] + a*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ 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