Internal problem ID [2011]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 11, page 45
Problem number: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {3 y^{2} y^{\prime }-x y^{3}={\mathrm e}^{\frac {x^{2}}{2}} \cos \left (x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 144
dsolve(3*y(x)^2*diff(y(x),x)-x*y(x)^3=exp(x^2/2)*cos(x),y(x), singsol=all)
\begin{align*} y = {\mathrm e}^{\frac {x^{2}}{2}} \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{\frac {1}{3}} y = -\frac {{\mathrm e}^{\frac {x^{2}}{2}} \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, {\mathrm e}^{\frac {x^{2}}{2}} \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{\frac {1}{3}}}{2} y = -\frac {{\mathrm e}^{\frac {x^{2}}{2}} \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, {\mathrm e}^{\frac {x^{2}}{2}} \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{\frac {1}{3}}}{2} \end{align*}
✓ Solution by Mathematica
Time used: 0.426 (sec). Leaf size: 81
DSolve[3*y[x]^2*y'[x]-x*y[x]^3==Exp[x^2/2]*Cos[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {x^2}{6}} \sqrt [3]{\sin (x)+c_1} y(x)\to -\sqrt [3]{-1} e^{\frac {x^2}{6}} \sqrt [3]{\sin (x)+c_1} y(x)\to (-1)^{2/3} e^{\frac {x^2}{6}} \sqrt [3]{\sin (x)+c_1} \end{align*}