78.2.41 problem 17.a

Internal problem ID [20993]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 17.a
Date solved : Thursday, October 02, 2025 at 07:01:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-k^{2} y&=f \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 53

ode:=diff(diff(y(x),x),x)-k^2*y(x) = f(x); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {-{\mathrm e}^{-k x} \left (1+\int _{0}^{x}{\mathrm e}^{k \textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} \right )+{\mathrm e}^{k x} \left (\int _{0}^{x}{\mathrm e}^{-k \textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} +1\right )}{2 k} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 144

ode=D[y[x],{x,2}]-k^2*y[x]==f[x]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-k x} \left (-2 k e^{2 k x} \int _1^0\frac {e^{-k K[1]} f(K[1])}{2 k}dK[1]+2 k e^{2 k x} \int _1^x\frac {e^{-k K[1]} f(K[1])}{2 k}dK[1]+2 k \int _1^x-\frac {e^{k K[2]} f(K[2])}{2 k}dK[2]-2 k \int _1^0-\frac {e^{k K[2]} f(K[2])}{2 k}dK[2]+e^{2 k x}-1\right )}{2 k} \end{align*}

✓ Sympy. Time used: 0.457 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(-k**2*y(x) - f(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {\int f{\left (x \right )} e^{- k x}\, dx}{2 k} - \frac {\int \limits ^{0} f{\left (x \right )} e^{- k x}\, dx}{2 k} + \frac {1}{2 k}\right ) e^{k x} + \left (- \frac {\int f{\left (x \right )} e^{k x}\, dx}{2 k} + \frac {\int \limits ^{0} f{\left (x \right )} e^{k x}\, dx}{2 k} - \frac {1}{2 k}\right ) e^{- k x} \]