60.1.402 problem 413
Internal
problem
ID
[10416]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
413
Date
solved
:
Wednesday, March 05, 2025 at 10:47:17 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} x {y^{\prime }}^{2}+y y^{\prime }-x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.081 (sec). Leaf size: 272
ode:=x*diff(y(x),x)^2+y(x)*diff(y(x),x)-x^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
-\int _{\textit {\_b}}^{x}\frac {y-\sqrt {4 \textit {\_a}^{3}+y^{2}}}{\textit {\_a} \left (4 y-\sqrt {4 \textit {\_a}^{3}+y^{2}}\right )}d \textit {\_a} +2 \left (\int _{}^{y}\frac {-1+\left (24 \textit {\_f} -6 \sqrt {4 x^{3}+\textit {\_f}^{2}}\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{4 \textit {\_f} -\sqrt {4 x^{3}+\textit {\_f}^{2}}}d \textit {\_f} \right )+c_{1} &= 0 \\
-\int _{\textit {\_b}}^{x}\frac {y+\sqrt {4 \textit {\_a}^{3}+y^{2}}}{\left (\sqrt {4 \textit {\_a}^{3}+y^{2}}+4 y\right ) \textit {\_a}}d \textit {\_a} -2 \left (\int _{}^{y}\frac {1+6 \left (\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right )}{\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f}}d \textit {\_f} \right )+c_{1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.261 (sec). Leaf size: 673
ode=-x^2 + y[x]*D[y[x],x] + x*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\int \left (\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x) \left (4 x^3-15 y(x)^2\right )}+\frac {16 x^2}{5 \left (4 x^3-15 y(x)^2\right )}-\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x) x}+\frac {1}{5 x}\right )dx+\int \left (\frac {8 y(x)}{15 y(x)^2-4 x^3}-\int \left (-\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x)^2 \left (4 x^3-15 y(x)^2\right )}+\frac {24 \sqrt {4 x^3+y(x)^2} x^2}{\left (4 x^3-15 y(x)^2\right )^2}+\frac {4 x^2}{5 \left (4 x^3-15 y(x)^2\right ) \sqrt {4 x^3+y(x)^2}}+\frac {96 y(x) x^2}{\left (4 x^3-15 y(x)^2\right )^2}+\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x)^2 x}-\frac {1}{5 \sqrt {4 x^3+y(x)^2} x}\right )dx+\frac {2 \sqrt {4 x^3+y(x)^2}}{15 y(x)^2-4 x^3}\right )dy(x)&=c_1,y(x)\right ] \\
\text {Solve}\left [\int \left (-\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x) \left (4 x^3-15 y(x)^2\right )}+\frac {16 x^2}{5 \left (4 x^3-15 y(x)^2\right )}+\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x) x}+\frac {1}{5 x}\right )dx+\int \left (\frac {8 y(x)}{15 y(x)^2-4 x^3}-\int \left (\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x)^2 \left (4 x^3-15 y(x)^2\right )}-\frac {24 \sqrt {4 x^3+y(x)^2} x^2}{\left (4 x^3-15 y(x)^2\right )^2}-\frac {4 x^2}{5 \left (4 x^3-15 y(x)^2\right ) \sqrt {4 x^3+y(x)^2}}+\frac {96 y(x) x^2}{\left (4 x^3-15 y(x)^2\right )^2}-\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x)^2 x}+\frac {1}{5 \sqrt {4 x^3+y(x)^2} x}\right )dx-\frac {2 \sqrt {4 x^3+y(x)^2}}{15 y(x)^2-4 x^3}\right )dy(x)&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 + x*Derivative(y(x), x)**2 + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(4*x**3 + y(x)**2) - y(x))/(2*x) cannot be solved by the factorable group method