15.7.1 problem 1

Internal problem ID [2982]
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
Date solved : Monday, January 27, 2025 at 07:05:30 AM
CAS classification : [_Bernoulli]

\begin{align*} 3 y^{2} y^{\prime }-x y^{3}&={\mathrm e}^{\frac {x^{2}}{2}} \cos \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 86

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 &= \left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{{1}/{3}} {\mathrm e}^{\frac {x^{2}}{2}} \\ y &= -\frac {\left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {x^{2}}{2}}}{2} \\ y &= \frac {\left (\left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{-x^{2}}\right )^{{1}/{3}} {\mathrm e}^{\frac {x^{2}}{2}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.488 (sec). Leaf size: 81

DSolve[3*y[x]^2*D[y[x],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*}