60.1.385 problem 386

Internal problem ID [10399]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 386
Date solved : Monday, January 27, 2025 at 07:39:05 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.424 (sec). Leaf size: 27

dsolve(diff(y(x),x)^2+a*x^3*diff(y(x),x)-2*a*x^2*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {a \,x^{4}}{8} \\ y &= \frac {c_{1} \left (a \,x^{2}+2 c_{1} \right )}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.462 (sec). Leaf size: 78

DSolve[-2*a*x^2*y[x] + a*x^3*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 8 e^{4 c_1}-2 \sqrt {a} e^{2 c_1} x^2 \\ y(x)\to \frac {1}{8} e^{2 c_1} \left (2 \sqrt {a} x^2+e^{2 c_1}\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}