73.7.6 problem 6

Internal problem ID [15164]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 07:38:56 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x y y^{\prime }-y^{2}&=\sqrt {x^{4}+x^{2} y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

dsolve(x*y(x)*diff(y(x),x)-y(x)^2=sqrt(x^4+x^2*y(x)^2),y(x), singsol=all)
 

\[ -\frac {x^{2}+y^{2}}{\sqrt {x^{2} \left (x^{2}+y^{2}\right )}}+\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.292 (sec). Leaf size: 54

DSolve[x*y[x]*D[y[x],x]-y[x]^2==Sqrt[x^4+x^2*y[x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x \sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ y(x)\to x \sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ \end{align*}