14.10.10 problem 10

Internal problem ID [2592]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.4. The method of variation of parameters. Excercises page 156
Problem number : 10
Date solved : Sunday, March 30, 2025 at 12:11:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+p \left (t \right ) y^{\prime }+q \left (t \right ) y&=1+t \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\left (1+t \right )^{2} \end{align*}

✗ Maple

ode:=diff(diff(y(t),t),t)+p(t)*diff(y(t),t)+q(t)*y(t) = t+1; 
dsolve(ode,y(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[t],{t,2}]+p[t]*D[y[t],t]+q[t]*y[t]==1+t; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(t - q(t)*y(t) - Derivative(y(t), (t, 2)) + 1)/p(t) + Derivative(y(t), t) cannot be solved by the factorable group method