78.23.6 problem 4

Internal problem ID [18457]
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 : Tuesday, January 28, 2025 at 11:50:21 AM
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*}

✓ Solution by Maple

Time used: 0.183 (sec). Leaf size: 22

dsolve([diff(y(x),x$2)-2*a*diff(y(x),x)+a^2*y(x)=0,y(0) = y__0, D(y)(0) = yd__0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 21

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

\[ y(x)\to e^{a x} (-a x \text {y0}+x \text {yd0}+\text {y0}) \]