76.3.4 problem 4

Internal problem ID [17375]
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 : 4
Date solved : Tuesday, January 28, 2025 at 10:03:30 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 (-3\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 55

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

\[ y = \frac {3 t}{2}+\frac {3 \ln \left (t +2\right ) t^{2}}{8}-\frac {3 \ln \left (t +2\right )}{2}-\frac {3 \ln \left (t -2\right ) t^{2}}{8}+\frac {3 \ln \left (t -2\right )}{2}+\frac {11 t^{2}}{10}-\frac {22}{5}+\frac {3 \ln \left (5\right ) t^{2}}{8}-\frac {3 \ln \left (5\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 39

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

\[ y(t)\to \frac {1}{5} \left (t^2-4\right ) \left (5 \int _{-3}^t-\frac {3 K[1]^2}{\left (K[1]^2-4\right )^2}dK[1]+1\right ) \]