76.17.25 problem 41

Internal problem ID [17711]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 41
Date solved : Tuesday, January 28, 2025 at 11:01:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 25

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

\[ y = c_{2} t +c_{1} {\mathrm e}^{t}-{\mathrm e}^{-t} t +\frac {{\mathrm e}^{-t}}{2} \]

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 30

DSolve[(1-t)*D[y[t],{t,2}]+t*D[y[t],t]-y[t]==2*(t-1)^2*Exp[-t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t} \left (\frac {1}{2}-t\right )+c_1 e^t-c_2 t \]