29.24.7 problem 669

Internal problem ID [5260]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 669
Date solved : Monday, January 27, 2025 at 10:50:32 AM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y\right )^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 40

dsolve((x^2+1)*(1+y(x)^2)*diff(y(x),x)+2*x*y(x)*(1-y(x))^2 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.346 (sec). Leaf size: 40

DSolve[(1+x^2)(1+y[x]^2)D[y[x],x]+2 x y[x](1-y[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\log (\text {$\#$1})-\frac {2}{\text {$\#$1}-1}\&\right ]\left [-\log \left (x^2+1\right )+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}