87.17.41 problem 50

Internal problem ID [23633]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 127
Problem number : 50
Date solved : Thursday, October 02, 2025 at 09:43:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+7 y^{\prime }+3 y&=5 \cos \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-3 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 25

ode:=4*diff(diff(y(t),t),t)+7*diff(y(t),t)+3*y(t) = 5*cos(t); 
ic:=[y(0) = -3, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {17 \,{\mathrm e}^{-t}}{2}+\frac {28 \,{\mathrm e}^{-\frac {3 t}{4}}}{5}-\frac {\cos \left (t \right )}{10}+\frac {7 \sin \left (t \right )}{10} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 41

ode=4*D[y[t],{t,2}]+7*D[y[y],t]+3*y[t]==5*Cos[t]; 
ic={y[0]==-3,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {10 \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}}-5 \cos (t)+2 \cos \left (\frac {\sqrt {3} t}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) - 5*cos(t) + 7*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {y(0): -3, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {7 \sin {\left (t \right )}}{10} - \frac {\cos {\left (t \right )}}{10} - \frac {17 e^{- t}}{2} + \frac {28 e^{- \frac {3 t}{4}}}{5} \]