40.4.23 problem 23 (c)

Internal problem ID [6663]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 6. Equations of first order and first degree (Linear equations). Supplemetary problems. Page 39
Problem number : 23 (c)
Date solved : Wednesday, March 05, 2025 at 01:36:22 AM
CAS classification : [_Bernoulli]

\begin{align*} x y^{3}-y^{3}-x^{2} {\mathrm e}^{x}+3 x y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 99

ode:=x*y(x)^3-y(x)^3-x^2*exp(x)+3*x*y(x)^2*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{2}/{3}} {\left (\left ({\mathrm e}^{2 x}+2 c_1 \right ) x \,{\mathrm e}^{2 x}\right )}^{{1}/{3}} {\mathrm e}^{-x}}{2} \\ y &= -\frac {2^{{2}/{3}} {\left (\left ({\mathrm e}^{2 x}+2 c_1 \right ) x \,{\mathrm e}^{2 x}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\mathrm e}^{-x}}{4} \\ y &= \frac {2^{{2}/{3}} {\left (\left ({\mathrm e}^{2 x}+2 c_1 \right ) x \,{\mathrm e}^{2 x}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\mathrm e}^{-x}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.903 (sec). Leaf size: 117

ode=(x*y[x]^3-y[x]^3-x^2*Exp[x])+(3*x*y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.265 (sec). Leaf size: 85

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

\[ \left [ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{x \left (C_{1} e^{- x} + e^{x}\right )}}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{x \left (C_{1} e^{- x} + e^{x}\right )} \left (-1 - \sqrt {3} i\right )}{4}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{x \left (C_{1} e^{- x} + e^{x}\right )} \left (-1 + \sqrt {3} i\right )}{4}\right ] \]