29.26.16 problem 752
Internal
problem
ID
[5337]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
752
Date
solved
:
Tuesday, March 04, 2025 at 09:29:39 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}+x^{2}&=4 y \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 136
ode:=diff(y(x),x)^2+x^2 = 4*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {x \sqrt {2}\, {\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )^{2}} \\
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (-\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}} \\
y \left (x \right ) &= \frac {x^{2} \left (2 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )+1\right )}{4 \operatorname {LambertW}\left (\frac {c_{1} \sqrt {2}\, x}{2}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 17.688 (sec). Leaf size: 152
ode=(D[y[x],x])^2+x^2==4*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [-\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )+\frac {x \sqrt {4 y(x)-x^2}}{2 x^2-4 y(x)}+\int \frac {x \left (x^2-4 y(x)\right )}{\left (x^2-2 y(x)\right )^2} \, dx&=c_1,y(x)\right ] \\
\text {Solve}\left [\text {arctanh}\left (\frac {x}{\sqrt {4 y(x)-x^2}}\right )-\frac {x \sqrt {4 y(x)-x^2}}{2 \left (x^2-2 y(x)\right )}+\int \frac {x \left (x^2-4 y(x)\right )}{\left (x^2-2 y(x)\right )^2} \, dx&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 - 4*y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-x**2 + 4*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method