69.33.16 problem 845

Internal problem ID [18585]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3. Section 24.2. Solving the Cauchy problem for linear differential equation with constant coefficients. Exercises page 249
Problem number : 845
Date solved : Thursday, October 02, 2025 at 03:15:06 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} 2 x^{\prime \prime }-2 x^{\prime }&=\left (t +1\right ) {\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&={\frac {1}{2}} \\ x^{\prime }\left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 13

ode:=2*diff(diff(x(t),t),t)-2*diff(x(t),t) = (t+1)*exp(t); 
ic:=[x(0) = 1/2, D(x)(0) = 1/2]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {{\mathrm e}^{t} \left (t^{2}+2\right )}{4} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 17

ode=2*D[x[t],{t,2}]-2*D[x[t],t]==(1+t)*Exp[t]; 
ic={x[0]==1/2,Derivative[1][x][0 ]==1/2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^t \left (t^2+2\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((-t - 1)*exp(t) - 2*Derivative(x(t), t) + 2*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1/2, Subs(Derivative(x(t), t), t, 0): 1/2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {t^{2}}{4} + \frac {1}{2}\right ) e^{t} \]