25.6.16 problem 16
Internal
problem
ID
[6900]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
16
Date
solved
:
Tuesday, September 30, 2025 at 04:01:19 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x -y y^{\prime }&=a {y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.120 (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 {\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 ) \sqrt {2}\, \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: 0.254 (sec). Leaf size: 129
ode=x-y[x]*D[y[x],x]==a*(D[y[x],x])^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=-a \exp \left (\int _1^{K[1]}-\frac {1}{K[2]^2 \left (K[2]-\frac {1}{K[2]}\right )}dK[2]\right ) \int \frac {\exp \left (-\int _1^{K[1]}-\frac {1}{K[2]^2 \left (K[2]-\frac {1}{K[2]}\right )}dK[2]\right )}{K[1]-\frac {1}{K[1]}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}-\frac {1}{K[2]^2 \left (K[2]-\frac {1}{K[2]}\right )}dK[2]\right ),y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ]
\]
✗ 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