68.14.33 problem 33

Internal problem ID [17725]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 33
Date solved : Thursday, October 02, 2025 at 02:27:23 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=1 \\ y^{\prime \prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 14

ode:=2*t^3*diff(diff(diff(y(t),t),t),t)+t^2*diff(diff(y(t),t),t)+t*diff(y(t),t)-y(t) = -3*t^2; 
ic:=[y(1) = 0, D(y)(1) = 1, (D@@2)(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -t \left (-2 \ln \left (t \right )+t -1\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 16

ode=2*t^3*D[ y[t],{t,3}]+t^2*D[y[t],{t,2}]+t*D[y[t],t]-y[t]==-3*t^2; 
ic={y[1]==0,Derivative[1][y][1]==1,Derivative[2][y][1]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t (-t+2 \log (t)+1) \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 14

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

\[ y{\left (t \right )} = - t^{2} + 2 t \log {\left (t \right )} + t \]