12.8.3 problem 2d

Internal problem ID [1739]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.1 Homogeneous linear equations. Page 203
Problem number : 2d
Date solved : Monday, January 27, 2025 at 05:33:59 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 20

dsolve([diff(y(x),x$2)-2*diff(y(x),x)+2*y(x)=0,y(0) = k__0, D(y)(0) = k__1],y(x), singsol=all)
 

\[ y = {\mathrm e}^{x} \left (\left (k_{1} -k_{0} \right ) \sin \left (x \right )+k_{0} \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 22

DSolve[{D[y[x],{x,2}]-2*D[y[x],x]+2*y[x]==0,{y[0]==k0,Derivative[1][y][0] ==k1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x ((\text {k1}-\text {k0}) \sin (x)+\text {k0} \cos (x)) \]