Internal problem ID [3536]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 805.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 2.078 (sec). Leaf size: 27
dsolve(diff(y(x),x)^2-2*a*x^3*diff(y(x),x)+4*a*x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {a \,x^{4}}{4} \\ y \relax (x ) = c_{1} x^{2}-\frac {c_{1}^{2}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.671 (sec). Leaf size: 406
DSolve[(y'[x])^2-2 a x^3 y'[x]+4 a x^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {a x^4-4 K[1]} \sqrt {a x^2 \left (a x^4-4 K[1]\right )}}{4 a^{3/2} x^5 K[1]}-\frac {\sqrt {a x^4-4 K[1]} \sqrt {a x^2 \left (a x^4-4 K[1]\right )}}{a^{3/2} x^5 \left (4 K[1]-a x^4\right )}+\frac {1}{4 K[1]}\right )dK[1]-\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 [\int _1^{y(x)}\left (-\frac {\sqrt {a x^4-4 K[2]} \sqrt {a x^2 \left (a x^4-4 K[2]\right )}}{4 a^{3/2} x^5 K[2]}+\frac {\sqrt {a x^4-4 K[2]} \sqrt {a x^2 \left (a x^4-4 K[2]\right )}}{a^{3/2} x^5 \left (4 K[2]-a x^4\right )}+\frac {1}{4 K[2]}\right )dK[2]+\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 ] \\ y(x)\to \frac {a x^4}{4} \\ \end{align*}