29.7.15 problem 190

Internal problem ID [4790]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 7
Problem number : 190
Date solved : Monday, January 27, 2025 at 09:38:02 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} x y^{\prime }&=a y+b \left (x^{2}+1\right ) y^{3} \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 150

dsolve(x*diff(y(x),x) = a*y(x)+b*(x^2+1)*y(x)^3,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 4.509 (sec). Leaf size: 108

DSolve[x D[y[x],x]==a y[x]+b(1+x^2)y[x]^3,y[x],x,IncludeSingularSolutions -> True]
 

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