14.9.14 problem 17

Internal problem ID [2580]
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 : 17
Date solved : Monday, January 27, 2025 at 06:01:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }-\left (1+3 t \right ) y^{\prime }+3 y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 16

dsolve([t*diff(y(t),t$2)-(1+3*t)*diff(y(t),t)+3*y(t)=0,exp(c*t)],singsol=all)
 

\[ y = c_2 \,{\mathrm e}^{3 t}+3 c_1 t +c_1 \]

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 25

DSolve[t*D[y[t],{t,2}]-(1+3*t)*D[y[t],t]+3*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to c_1 e^{3 t}-\frac {1}{9} c_2 (3 t+1) \]