81.3.23 problem 23

Internal problem ID [18641]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 23
Date solved : Tuesday, January 28, 2025 at 12:06:33 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 5 x y y^{\prime }-x^{2}-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 37

dsolve(5*x*y(x)*diff(y(x),x)-(x^2+y(x)^2)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {x^{{2}/{5}} \left (x^{{8}/{5}}+4 c_{1} \right )}}{2} \\ y \left (x \right ) &= \frac {\sqrt {x^{{2}/{5}} \left (x^{{8}/{5}}+4 c_{1} \right )}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.490 (sec). Leaf size: 50

DSolve[5*x*y[x]*D[y[x],x]-(x^2+y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {x^2+4 c_1 x^{2/5}} \\ y(x)\to \sqrt {\frac {x^2}{4}+c_1 x^{2/5}} \\ \end{align*}