20.9.23 problem Problem 23

Internal problem ID [3767]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 23
Date solved : Sunday, March 30, 2025 at 02:07:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-9 y&=F \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x)-9*y(x) = F(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-3 x} \left (\left (\int {\mathrm e}^{-3 x} F \left (x \right )d x +6 c_2 \right ) {\mathrm e}^{6 x}-\int {\mathrm e}^{3 x} F \left (x \right )d x +6 c_1 \right )}{6} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 66

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

\[ y(x)\to e^{-x} \left (e^{2 x} \int _1^x\frac {1}{2} e^{-K[1]} F(K[1])dK[1]+\int _1^x-\frac {1}{2} e^{K[2]} F(K[2])dK[2]+c_1 e^{2 x}+c_2\right ) \]

✓ Sympy. Time used: 0.690 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-F(x) - 9*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {\int F{\left (x \right )} e^{3 x}\, dx}{6}\right ) e^{- 3 x} + \left (C_{2} + \frac {\int F{\left (x \right )} e^{- 3 x}\, dx}{6}\right ) e^{3 x} \]