7.20.4 problem 32

Internal problem ID [558]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.4 (Derivatives, Integrals and products of transforms). Problems at page 303
Problem number : 32
Date solved : Monday, January 27, 2025 at 02:54:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.215 (sec). Leaf size: 20

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

\[ x \left (t \right ) = \frac {{\mathrm e}^{-t} \left (t \cosh \left (t \right )-\sinh \left (t \right )\right ) c_1}{2} \]

✓ Solution by Mathematica

Time used: 0.142 (sec). Leaf size: 46

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

\[ x(t)\to -\frac {\sqrt {\frac {2}{\pi }} c_1 e^{-t} \sqrt {t} (t \cosh (t)-\sinh (t))}{\sqrt {-i t}} \]