74.18.45 problem 51

Internal problem ID [16602]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 51
Date solved : Tuesday, January 28, 2025 at 09:12:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.080 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{t} \left (2 \ln \left (t \right ) t^{2}-4 \,{\mathrm e}^{-1} t -3 t^{2}+8 \,{\mathrm e}^{-1}+4 t -1\right )}{4} \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 39

DSolve[{D[y[t],{t,2}]-2*D[y[t],t]+y[t]==Exp[t]*Log[t],{y[1]==1,Derivative[1][y][1]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{4} e^{t-1} \left (e \left (-3 t^2+4 t-1\right )+2 e t^2 \log (t)-4 t+8\right ) \]