23.1.230 problem 226

Internal problem ID [4837]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 226
Date solved : Tuesday, September 30, 2025 at 08:43:46 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: 0.047 (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]
 

\begin{align*} y(x)&\to \left (e^x+c_1\right ) (x+1)^n \end{align*}

✓ Sympy. Time used: 8.818 (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)
 

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