6.8.3 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 : Tuesday, September 30, 2025 at 05:18:52 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*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(0) = k__0, D(y)(0) = k__1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 22

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

\begin{align*} y(x)&\to e^x ((\text {k1}-\text {k0}) \sin (x)+\text {k0} \cos (x)) \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): k__0, Subs(Derivative(y(x), x), x, 0): k__1} 
dsolve(ode,func=y(x),ics=ics)
 

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