76.33.29 problem Ex. 29

Internal problem ID [20207]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 29
Date solved : Thursday, October 02, 2025 at 05:34:40 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+y&={\mathrm e}^{2 x} \sin \left (x \right )+{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 67

ode:=diff(diff(diff(y(x),x),x),x)+y(x) = exp(2*x)*sin(x)+exp(1/2*x)*sin(1/2*3^(1/2)*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{\frac {x}{2}} \left (-6 c_3 +x -2\right ) \sin \left (\frac {x \sqrt {3}}{2}\right )}{6}-\frac {{\mathrm e}^{\frac {x}{2}} \left (x \sqrt {3}-6 c_2 \right ) \cos \left (\frac {x \sqrt {3}}{2}\right )}{6}+\frac {\left (-33 \cos \left (x \right )+9 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{390}+c_1 \,{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.488 (sec). Leaf size: 136

ode=D[y[x],{x,3}]+y[x]==Exp[2*x]*Sin[x]+Exp[x/2]*Sin[x*Sqrt[3]/2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {3}{130} e^{2 x} \sin (x)+\frac {1}{3} e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )-\frac {1}{6} e^{x/2} x \sin \left (\frac {\sqrt {3} x}{2}\right )-\frac {11}{130} e^{2 x} \cos (x)+c_1 e^{-x}-\frac {1}{6} e^{x/2} \left (\sqrt {3} x-6 c_3\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+c_2 e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.226 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = C_{3} e^{- x} + \left (\left (C_{1} - \frac {x}{6}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} - \frac {\sqrt {3} x}{6}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} + \frac {\left (3 \sin {\left (x \right )} - 11 \cos {\left (x \right )}\right ) e^{2 x}}{130} \]