68.18.45 problem 51

Internal problem ID [17889]
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 : Thursday, October 02, 2025 at 02:29:16 PM
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*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = exp(t)*ln(t); 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 39

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 37

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

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