14.9.8 problem 11

Internal problem ID [2574]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 11
Date solved : Monday, January 27, 2025 at 06:01:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{t^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 14

dsolve([diff(y(t),t$2)-4*t*diff(y(t),t)+(4*t^2-2)*y(t)=0,exp(t^2)],singsol=all)
 

\[ y = {\mathrm e}^{t^{2}} \left (c_2 t +c_1 \right ) \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 18

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

\[ y(t)\to e^{t^2} (c_2 t+c_1) \]