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 : Monday, January 27, 2025 at 11:17:46 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*}

✓ Solution by Maple

Time used: 0.509 (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 \left (x \right ) &= \frac {a \,x^{4}}{4} \\ y \left (x \right ) &= \frac {c_{1} \left (a \,x^{2}-c_{1} \right )}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.367 (sec). Leaf size: 382

DSolve[(D[y[x],x])^2-2*a*x^3*D[y[x],x]+4*a*x^2*y[x]==0,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*}