29.24.5 problem 667

Internal problem ID [5258]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 667
Date solved : Monday, January 27, 2025 at 10:50:06 AM
CAS classification : [_separable]

\begin{align*} x^{2} \left (a +y\right )^{2} y^{\prime }&=\left (x^{2}+1\right ) \left (y^{2}+a^{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 92

dsolve(x^2*(a+y(x))^2*diff(y(x),x) = (x^2+1)*(y(x)^2+a^2),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-a x \operatorname {RootOf}\left (\textit {\_Z}^{2} a^{2} x^{2}-2 c_{1} \textit {\_Z} a \,x^{2}-2 \textit {\_Z} a \,x^{3}+c_{1}^{2} x^{2}+2 c_{1} x^{3}+a^{2} x^{2}+x^{4}-x^{2} {\mathrm e}^{\textit {\_Z}}+2 a x \textit {\_Z} -2 c_{1} x -2 x^{2}+1\right )+c_{1} x +x^{2}-1}{x} \]

✓ Solution by Mathematica

Time used: 0.567 (sec). Leaf size: 48

DSolve[x^2 (a+y[x])^2 D[y[x],x]==(1+x^2)(a^2+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [a \log \left (\text {$\#$1}^2+a^2\right )+\text {$\#$1}\&\right ]\left [x-\frac {1}{x}+c_1\right ] \\ y(x)\to -i a \\ y(x)\to i a \\ \end{align*}