90.20.32 problem 19

Internal problem ID [25327]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 337
Problem number : 19
Date solved : Friday, October 03, 2025 at 12:00:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.188 (sec). Leaf size: 58

ode:=t^2*diff(diff(y(t),t),t)+3*t*diff(y(t),t)-4*y(t) = t^4; 
ic:=[y(-1) = y__1, D(y)(-1) = y__1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\left (y_{1} +\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) \left (-1\right )^{\sqrt {5}} t^{-\sqrt {5}-1}}{2}-\frac {\left (-1\right )^{-\sqrt {5}} \left (y_{1} -\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) t^{\sqrt {5}-1}}{2}+\frac {t^{4}}{20} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 81

ode=t^2*D[y[t],{t,2}]+3*t*D[y[t],t]-4*y[t]==t^4; 
ic={y[-1]==y1,Derivative[1][y][-1] ==y1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {-e^{i \sqrt {5} \pi } t^{-\sqrt {5}} \left (20 \text {y1}+\sqrt {5}-1\right )+e^{-i \sqrt {5} \pi } t^{\sqrt {5}} \left (-20 \text {y1}+\sqrt {5}+1\right )+2 t^5}{40 t} \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 100

from sympy import * 
t = symbols("t") 
y1 = symbols("y1") 
y = Function("y") 
ode = Eq(-t**4 + t**2*Derivative(y(t), (t, 2)) + 3*t*Derivative(y(t), t) - 4*y(t),0) 
ics = {y(-1): y1, Subs(Derivative(y(t), t), t, -1): y1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{4}}{20} + t^{-1 + \sqrt {5}} \left (- \frac {y_{1}}{2 \left (-1\right )^{\sqrt {5}}} + \frac {\sqrt {5}}{40 \left (-1\right )^{\sqrt {5}}} + \frac {1}{40 \left (-1\right )^{\sqrt {5}}}\right ) + \frac {- \frac {\left (-1\right )^{\sqrt {5}} y_{1}}{2} - \frac {\left (-1\right )^{\sqrt {5}} \sqrt {5}}{40} + \frac {\left (-1\right )^{\sqrt {5}}}{40}}{t t^{\sqrt {5}}} \]