69.1.43 problem 62

Internal problem ID [14117]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 62
Date solved : Wednesday, March 05, 2025 at 10:35:34 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }-\frac {n y}{x}&={\mathrm e}^{x} x^{n} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 12

ode:=diff(y(x),x)-n/x*y(x) = exp(x)*x^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_{1} +{\mathrm e}^{x}\right ) x^{n} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 15

ode=D[y[x],x]-n/x*y[x]==Exp[x]*x^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (e^x+c_1\right ) x^n \]

✓ Sympy. Time used: 5.750 (sec). Leaf size: 82

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

\[ \left [ y{\left (x \right )} = \begin {cases} \frac {C_{1} x e^{n \log {\left (x \right )}}}{n x e^{n \log {\left (x \right )}} \log {\left (x \right )} + 2 x} + \frac {x^{n + 1} e^{x}}{n x e^{n \log {\left (x \right )}} \log {\left (x \right )} + 2 x} & \text {for}\: n = 0 \vee n \geq \infty \vee n \leq -\infty \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{n \log {\left (x \right )}} + x^{n} e^{x} & \text {for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]