76.3.5 problem 5

Internal problem ID [17376]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 10:03:33 AM
CAS classification : [_linear]

\begin{align*} \left (-t^{2}+4\right ) y^{\prime }+2 t y&=3 t^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-3 \end{align*}

✓ Solution by Maple

Time used: 0.125 (sec). Leaf size: 62

dsolve([(4-t^2)*diff(y(t),t)+2*t*y(t)=3*t^2,y(1) = -3],y(t), singsol=all)
 

\[ y = \frac {\left (3 t^{2}-12\right ) \ln \left (t +2\right )}{8}+\frac {3 i \pi \,t^{2}}{8}-\frac {3 \ln \left (3\right ) t^{2}}{8}-\frac {3 \ln \left (t -2\right ) t^{2}}{8}-\frac {3 i \pi }{2}+\frac {3 t^{2}}{2}+\frac {3 t}{2}+\frac {3 \ln \left (3\right )}{2}+\frac {3 \ln \left (t -2\right )}{2}-6 \]

✓ Solution by Mathematica

Time used: 0.066 (sec). Leaf size: 34

DSolve[{(4-t^2)*D[y[t],t]+2*t*y[t]==3*t^2,{y[1]==-3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \left (t^2-4\right ) \left (\int _1^t-\frac {3 K[1]^2}{\left (K[1]^2-4\right )^2}dK[1]+1\right ) \]