15.14.31 problem 33

Internal problem ID [3203]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 33
Date solved : Tuesday, March 04, 2025 at 04:05:51 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 46

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

\[ y = -\frac {c_2 \sqrt {2}\, \cos \left (\sqrt {2}\, x \right )}{2}+\frac {\sqrt {2}\, \sin \left (\sqrt {2}\, x \right ) c_{1}}{2}-x^{2} \cos \left (x \right )+8 \cos \left (x \right )-2 x \sin \left (x \right )+c_3 \]

✓ Mathematica. Time used: 0.168 (sec). Leaf size: 55

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

\[ y(x)\to -\left (x^2-8\right ) \cos (x)-2 x \sin (x)-\frac {c_2 \cos \left (\sqrt {2} x\right )}{\sqrt {2}}+\frac {c_1 \sin \left (\sqrt {2} x\right )}{\sqrt {2}}+c_3 \]

✓ Sympy. Time used: 0.258 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{1} + C_{2} \sin {\left (\sqrt {2} x \right )} + C_{3} \cos {\left (\sqrt {2} x \right )} - x^{2} \cos {\left (x \right )} - 2 x \sin {\left (x \right )} + 8 \cos {\left (x \right )} \]