75.33.16 problem 845

Internal problem ID [17298]
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 : Tuesday, January 28, 2025 at 09:59:01 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} 2 x^{\prime \prime }-2 x^{\prime }&=\left (1+t \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*}

✓ Solution by Maple

Time used: 9.763 (sec). Leaf size: 13

dsolve([2*diff(x(t),t$2)-2*diff(x(t),t)=(1+t)*exp(t),x(0) = 1/2, D(x)(0) = 1/2],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.063 (sec). Leaf size: 17

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

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