76.3.5 problem 5

Internal problem ID [17305]
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 : Monday, March 31, 2025 at 03:50:15 PM
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*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 62

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

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

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

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

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

✓ Sympy. Time used: 0.418 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t**2 + 2*t*y(t) + (4 - t**2)*Derivative(y(t), t),0) 
ics = {y(1): -3} 
dsolve(ode,func=y(t),ics=ics)
 

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