72.25.1 problem 5

Internal problem ID [19747]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 52. Convolutions and Abels Mechanical Problem. Problems at page 474
Problem number : 5
Date solved : Thursday, October 02, 2025 at 04:41:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.197 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)+a^2*y(x) = f(x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

\[ y = -\frac {\int _{0}^{x}f \left (\textit {\_U1} \right ) \sin \left (a \left (-x +\textit {\_U1} \right )\right )d \textit {\_U1}}{a} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 103

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

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

✓ Sympy. Time used: 0.526 (sec). Leaf size: 83

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

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