Internal problem ID [3530]
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]
\[ \boxed {x^{2} y^{\prime }-\left (x a +b y^{3}\right ) y=0} \]
✓ 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 \left (x \right ) = \frac {{\left (x \left (3 a -1\right ) \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )^{2}\right )}^{\frac {1}{3}}}{3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b} y \left (x \right ) = -\frac {{\left (x \left (3 a -1\right ) \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )}-\frac {i \sqrt {3}\, {\left (x \left (3 a -1\right ) \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )} y \left (x \right ) = -\frac {{\left (x \left (3 a -1\right ) \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )^{2}\right )}^{\frac {1}{3}}}{2 \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )}+\frac {i \sqrt {3}\, {\left (x \left (3 a -1\right ) \left (3 a \,x^{-3 a +1} c_{1} -x^{-3 a +1} c_{1} -3 b \right )^{2}\right )}^{\frac {1}{3}}}{6 a \,x^{-3 a +1} c_{1} -2 x^{-3 a +1} c_{1} -6 b} \end{align*}
✓ Solution by Mathematica
Time used: 3.526 (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*}