29.8.18 problem 223
Internal
problem
ID
[4823]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
8
Problem
number
:
223
Date
solved
:
Sunday, March 30, 2025 at 04:02:21 AM
CAS
classification
:
[_linear]
\begin{align*} \left (1+x \right ) y^{\prime }&={\mathrm e}^{x} \left (1+x \right )^{n +1}+n y \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 14
ode:=(1+x)*diff(y(x),x) = exp(x)*(1+x)^(n+1)+n*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \left ({\mathrm e}^{x}+c_1 \right ) \left (1+x \right )^{n}
\]
✓ Mathematica. Time used: 0.083 (sec). Leaf size: 17
ode=(1+x) D[y[x],x]==Exp[x](1+x)^(n+1)+n y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \left (e^x+c_1\right ) (x+1)^n
\]
✓ Sympy. Time used: 12.800 (sec). Leaf size: 196
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(-n*y(x) + (x + 1)*Derivative(y(x), x) - (x + 1)**(n + 1)*exp(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} \frac {C_{1} x e^{n \log {\left (x + 1 \right )}}}{x + 1} + \frac {C_{1} e^{n \log {\left (x + 1 \right )}}}{x + 1} + \left (x + 1\right )^{n} e^{x} & \text {for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1} x e^{n \log {\left (x + 1 \right )}}}{n x e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + n e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + 2 x + 2} + \frac {C_{1} e^{n \log {\left (x + 1 \right )}}}{n x e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + n e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + 2 x + 2} + \frac {\left (x + 1\right )^{n + 1} e^{x}}{n x e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + n e^{n \log {\left (x + 1 \right )}} \log {\left (x + 1 \right )} + 2 x + 2} & \text {for}\: n = 0 \vee n \geq \infty \vee n \leq -\infty \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]