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 : Sunday, March 30, 2025 at 01:03:19 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*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 86

ode:=3*y(x)^2*diff(y(x),x)-x*y(x)^3 = exp(1/2*x^2)*cos(x); 
dsolve(ode,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}} {\mathrm e}^{\frac {x^{2}}{2}} \left (1+i \sqrt {3}\right )}{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 (-1+i \sqrt {3}\right )}{2} \\ \end{align*}

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

ode=3*y[x]^2*D[y[x],x]-x*y[x]^3==Exp[x^2/2]*Cos[x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 11.701 (sec). Leaf size: 99

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x)**3 + 3*y(x)**2*Derivative(y(x), x) - exp(x**2/2)*cos(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt [3]{C_{1} e^{\frac {x^{2}}{2}} + \sqrt {e^{x^{2}}} \sin {\left (x \right )}}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} e^{\frac {x^{2}}{2}} + \sqrt {e^{x^{2}}} \sin {\left (x \right )}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} e^{\frac {x^{2}}{2}} + \sqrt {e^{x^{2}}} \sin {\left (x \right )}}}{2}\right ] \]