14.9.7 problem 10

Internal problem ID [2573]
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 : 10
Date solved : Monday, January 27, 2025 at 06:01:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {2 \left (1+t \right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1}&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=1+t \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 15

dsolve([diff(y(t),t$2)-2*(t+1)/(t^2+2*t-1)*diff(y(t),t)+2/(t^2+2*t-1)*y(t)=0,1+t],singsol=all)
 

\[ y = c_2 \,t^{2}+c_1 t +c_1 +c_2 \]

✓ Solution by Mathematica

Time used: 0.200 (sec). Leaf size: 64

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

\[ y(t)\to \frac {\sqrt {t^2+2 t-1} \left (c_1 \left (t^2-2 \left (\sqrt {2}-1\right ) t-2 \sqrt {2}+3\right )+c_2 (t+1)\right )}{\sqrt {-t^2-2 t+1}} \]