29.26.15 problem 751
Internal
problem
ID
[5336]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
751
Date
solved
:
Tuesday, March 04, 2025 at 09:29:37 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}&=x^{2}+y \end{align*}
✓ Maple. Time used: 0.051 (sec). Leaf size: 271
ode:=diff(y(x),x)^2 = y(x)+x^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
17 \ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )+17 \ln \left (-\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )-17 \ln \left (\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {y \left (x \right )+x^{2}}\right ) \sqrt {17}}{17 x}\right )+2 \,\operatorname {arctanh}\left (\frac {\left (x^{2}-8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {y \left (x \right )+x^{2}}+x \right ) \sqrt {17}}{17 x}\right )\right ) \sqrt {17}-c_{1} &= 0 \\
-17 \ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )+17 \ln \left (-\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )-17 \ln \left (\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {y \left (x \right )+x^{2}}\right ) \sqrt {17}}{17 x}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (x^{2}-8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {y \left (x \right )+x^{2}}+x \right ) \sqrt {17}}{17 x}\right )\right ) \sqrt {17}-c_{1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.263 (sec). Leaf size: 215
ode=(D[y[x],x])^2==x^2+y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {x^2+y(x)}-x\right )-\left (\sqrt {17}-17\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2-\sqrt {17} y(x)+3 y(x)\right )+\left (17+\sqrt {17}\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2+\left (3+\sqrt {17}\right ) y(x)\right )\right )&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {x^2+y(x)}-x\right )+\left (17+\sqrt {17}\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2+\left (\sqrt {17}-5\right ) y(x)\right )-\left (\sqrt {17}-17\right ) \log \left (2 x \sqrt {x^2+y(x)}-2 x^2-\left (5+\sqrt {17}\right ) y(x)\right )\right )&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 - y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(x**2 + y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method