18.2.13 problem Problem 15.24(a)

Internal problem ID [3496]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.24(a)
Date solved : Sunday, March 30, 2025 at 01:44:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 83

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

\[ y = \frac {x^{\frac {n}{2}} {\mathrm e}^{\frac {x}{2}} \operatorname {WhittakerM}\left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, x\right )-\left (n +1\right ) \left (x^{n} {\mathrm e}^{-x} \left (\Gamma \left (n , -x \right ) n -\Gamma \left (n +1\right )\right ) \left (-x \right )^{-n}-2 \,{\mathrm e}^{-x} c_1 -2 \,{\mathrm e}^{x} c_2 +x^{n}\right )}{2 n +2} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 58

ode=D[y[x],{x,2}]-y[x]==x^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {1}{2} e^{-x} x^n (-x)^{-n} \Gamma (n+1,-x)-\frac {1}{2} e^x \Gamma (n+1,x)+c_1 e^x+c_2 e^{-x} \]

✓ Sympy. Time used: 0.835 (sec). Leaf size: 107

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {n \Gamma \left (n + 1\right ) \gamma \left (n + 1, x\right )}{2 \Gamma \left (n + 2\right )} + \frac {\Gamma \left (n + 1\right ) \gamma \left (n + 1, x\right )}{2 \Gamma \left (n + 2\right )}\right ) e^{x} + \left (C_{2} + \frac {n e^{- i \pi n} \Gamma \left (n + 1\right ) \gamma \left (n + 1, x e^{i \pi }\right )}{2 \Gamma \left (n + 2\right )} + \frac {e^{- i \pi n} \Gamma \left (n + 1\right ) \gamma \left (n + 1, x e^{i \pi }\right )}{2 \Gamma \left (n + 2\right )}\right ) e^{- x} \]