14.10.11 problem 12

Internal problem ID [2593]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.4. The method of variation of parameters. Excercises page 156
Problem number : 12
Date solved : Monday, January 27, 2025 at 06:01:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {2 t y^{\prime }}{t^{2}+1}+\frac {2 y}{t^{2}+1}&=t^{2}+1 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 22

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

\[ y = c_2 t +c_1 \,t^{2}-c_1 +\frac {1}{2}+\frac {1}{6} t^{4} \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 33

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

\[ y(t)\to \frac {1}{6} \left (t^2+3\right ) t^2+c_2 t-c_1 (t-i)^2 \]