10.8 problem 274

Internal problem ID [3022]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 10
Problem number: 274.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class G], _rational, _Bernoulli]

Solve \begin {gather*} \boxed {y^{\prime } x^{2}-\left (a x +y^{3} b \right ) y=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 344

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

\begin{align*} y \relax (x ) = \frac {\left (x \left (3 a -1\right ) \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b} \\ y \relax (x ) = -\frac {\left (x \left (3 a -1\right ) \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )}-\frac {i \sqrt {3}\, \left (x \left (3 a -1\right ) \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )} \\ y \relax (x ) = -\frac {\left (x \left (3 a -1\right ) \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{2 \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )}+\frac {i \sqrt {3}\, \left (x \left (3 a -1\right ) \left (3 c_{1} a \,x^{-3 a +1}-c_{1} x^{-3 a +1}-3 b \right )^{2}\right )^{\frac {1}{3}}}{6 c_{1} a \,x^{-3 a +1}-2 c_{1} x^{-3 a +1}-6 b} \\ \end{align*}

Solution by Mathematica

Time used: 3.513 (sec). Leaf size: 149

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

\begin{align*} y(x)\to \frac {\sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{(1-3 a) x^{3 a+1}}}{\sqrt [3]{3 b x^{3 a}+(1-3 a) c_1 x}} \\ y(x)\to 0 \\ \end{align*}