82.6.4 problem Ex. 4

Internal problem ID [18666]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 22
Problem number : Ex. 4
Date solved : Thursday, March 13, 2025 at 12:31:45 PM
CAS classification : [[_homogeneous, `class D`], _Bernoulli]

\begin{align*} x^{4} {\mathrm e}^{x}-2 m x y^{2}+2 m \,x^{2} y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 42

ode:=x^4*exp(x)-2*m*x*y(x)^2+2*m*x^2*y(x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= \frac {\sqrt {m \left (c_{1} m -{\mathrm e}^{x}\right )}\, x}{m} \\ y \left (x \right ) &= -\frac {\sqrt {m \left (c_{1} m -{\mathrm e}^{x}\right )}\, x}{m} \\ \end{align*}

✓ Mathematica. Time used: 7.767 (sec). Leaf size: 58

ode=(x^4*Exp[x]-2*m*x*y[x]^2)+2*m*x^2*y[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-x^2 \left (e^x-c_1 m\right )}}{\sqrt {m}} \\ y(x)\to \frac {\sqrt {-x^2 \left (e^x-c_1 m\right )}}{\sqrt {m}} \\ \end{align*}

✓ Sympy. Time used: 0.752 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
m = symbols("m") 
y = Function("y") 
ode = Eq(2*m*x**2*y(x)*Derivative(y(x), x) - 2*m*x*y(x)**2 + x**4*exp(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \begin {cases} - x \sqrt {C_{1} - \frac {e^{x}}{m}} & \text {for}\: m > 0 \vee m < 0 \\- x \sqrt {C_{1} - \frac {x}{m}} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} x \sqrt {C_{1} - \frac {e^{x}}{m}} & \text {for}\: m > 0 \vee m < 0 \\x \sqrt {C_{1} - \frac {x}{m}} & \text {otherwise} \end {cases}\right ] \]