72.23.6 problem 4

Internal problem ID [19743]
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 50. Applications to differential equations. Problems at page 462
Problem number : 4
Date solved : Thursday, October 02, 2025 at 04:41:52 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 a y^{\prime }+a^{2} y&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.094 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-2*a*diff(y(x),x)+a^2*y(x) = 0; 
ic:=[y(0) = y__0, D(y)(0) = yd__0]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

\[ y = {\mathrm e}^{a x} \left (-a x y_{0} +x \operatorname {yd}_{0} +y_{0} \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 21

ode=D[y[x],{x,2}]-2*a*D[y[x],x]+a^2*y[x]==0; 
ic={y[0]==y0,Derivative[1][y][0] == yd0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{a x} (-a x \text {y0}+x \text {yd0}+\text {y0}) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*y(x) - 2*a*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): y__0, Subs(Derivative(y(x), x), x, 0): yd__0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x \left (- a y^{0} + yd^{0}\right ) + y^{0}\right ) e^{a x} \]