9.11.33 problem 33

Internal problem ID [3143]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 33
Date solved : Tuesday, September 30, 2025 at 06:27:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 y^{\prime \prime }+5 y^{\prime }-3 y&=\sin \left (x \right )-8 x \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \\ y^{\prime }\left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 29

ode:=2*diff(diff(y(x),x),x)+5*diff(y(x),x)-3*y(x) = sin(x)-8*x; 
ic:=[y(0) = 1/2, D(y)(0) = 1/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {136 \,{\mathrm e}^{\frac {x}{2}}}{35}+\frac {13 \,{\mathrm e}^{-3 x}}{315}+\frac {8 x}{3}+\frac {40}{9}-\frac {\cos \left (x \right )}{10}-\frac {\sin \left (x \right )}{10} \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 38

ode=2*D[y[x],{x,2}]+5*D[y[x],x]-3*y[x]==Sin[x]-8*x; 
ic={y[0]==1/2,Derivative[1][y][0] ==1/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{630} \left (1680 x+26 e^{-3 x}-2448 e^{x/2}-63 \sin (x)-63 \cos (x)+2800\right ) \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x - 3*y(x) - sin(x) + 5*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1/2, Subs(Derivative(y(x), x), x, 0): 1/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {8 x}{3} - \frac {136 e^{\frac {x}{2}}}{35} - \frac {\sin {\left (x \right )}}{10} - \frac {\cos {\left (x \right )}}{10} + \frac {40}{9} + \frac {13 e^{- 3 x}}{315} \]