7.20.6 problem 34

Internal problem ID [560]
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 : 34
Date solved : Monday, January 27, 2025 at 02:54:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t x^{\prime \prime }+\left (4 t -2\right ) x^{\prime }+\left (13 t -4\right ) x&=0 \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.206 (sec). Leaf size: 24

dsolve([t*diff(x(t),t$2)+(4*t-2)*diff(x(t),t)+(13*t-4)*x(t)=0,x(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = -\frac {{\mathrm e}^{-2 t} c_1 \left (\cos \left (3 t \right ) t -\frac {\sin \left (3 t \right )}{3}\right )}{18} \]

✓ Solution by Mathematica

Time used: 0.167 (sec). Leaf size: 41

DSolve[{t*D[x[t],{t,2}]+(4*t-2)*D[x[t],t]+(13*t-4)*x[t]==0,{x[0]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\frac {1}{3} \sqrt {\frac {2}{3 \pi }} c_1 e^{-2 t} (3 t \cos (3 t)-\sin (3 t)) \]