29.28.7 problem 805
Internal
problem
ID
[5388]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
28
Problem
number
:
805
Date
solved
:
Sunday, March 30, 2025 at 08:04:16 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} {y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y&=0 \end{align*}
✓ Maple. Time used: 0.363 (sec). Leaf size: 27
ode:=diff(y(x),x)^2-2*a*x^3*diff(y(x),x)+4*a*x^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {a \,x^{4}}{4} \\
y &= \frac {c_{1} \left (a \,x^{2}-c_{1} \right )}{a} \\
\end{align*}
✓ Mathematica. Time used: 5.367 (sec). Leaf size: 382
ode=(D[y[x],x])^2-2*a*x^3*D[y[x],x]+4*a*x^2*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {1}{4} \left (\frac {\sqrt {a} x \sqrt {a x^4-4 y(x)}}{\sqrt {a x^2 \left (a x^4-4 y(x)\right )}}+1\right ) \log (-4 y(x))-\frac {\sqrt {a} x \sqrt {a x^4-4 y(x)} \log \left (\sqrt {a x^4-4 y(x)}-\sqrt {a} x^2\right )}{2 \sqrt {a x^2 \left (a x^4-4 y(x)\right )}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\sqrt {a} x \sqrt {a x^4-4 y(x)} \log \left (\sqrt {a x^4-4 y(x)}-\sqrt {a} x^2\right )}{2 \sqrt {a x^2 \left (a x^4-4 y(x)\right )}}+\frac {\left (\sqrt {a x^2}+\sqrt {a} x\right ) \left (\log \left (-a x^3-\frac {\left (a x^2\right )^{3/2}}{\sqrt {a}}-\sqrt {a x^2} \sqrt {a x^4-4 y(x)}+\sqrt {a x^2 \left (a x^4-4 y(x)\right )}\right )+\log \left (3 a x^3-\frac {\left (a x^2\right )^{3/2}}{\sqrt {a}}-\sqrt {a x^2} \sqrt {a x^4-4 y(x)}+\sqrt {a x^2 \left (a x^4-4 y(x)\right )}\right )\right )}{4 \sqrt {a} x}&=c_1,y(x)\right ] \\
y(x)\to \frac {a x^4}{4} \\
\end{align*}
✓ Sympy. Time used: 2.232 (sec). Leaf size: 12
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-2*a*x**3*Derivative(y(x), x) + 4*a*x**2*y(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} a \left (- C_{1} - x^{2}\right )
\]