15.15.13 problem 14

Internal problem ID [3217]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 24, page 109
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:21:28 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 54

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

\[ y = \frac {\left (-8 x^{3}+24 x^{2}+6 x -15\right ) \sin \left (2 x \right )}{64}+\frac {\left (-4 x^{3}-6 x^{2}+15 x \right ) \cos \left (2 x \right )}{32}-\frac {{\mathrm e}^{-2 x} c_1}{2}+c_2 \]

✓ Mathematica. Time used: 0.502 (sec). Leaf size: 61

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

\[ y(x)\to -\frac {1}{32} x \left (4 x^2+6 x-15\right ) \cos (2 x)+\frac {1}{64} \left (-8 x^3+24 x^2+6 x-15\right ) \sin (2 x)-\frac {1}{2} c_1 e^{-2 x}+c_2 \]

✓ Sympy. Time used: 0.390 (sec). Leaf size: 83

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

\[ y{\left (x \right )} = C_{1} + C_{2} e^{- 2 x} - \frac {x^{3} \sin {\left (2 x \right )}}{8} - \frac {x^{3} \cos {\left (2 x \right )}}{8} + \frac {3 x^{2} \sin {\left (2 x \right )}}{8} - \frac {3 x^{2} \cos {\left (2 x \right )}}{16} + \frac {3 x \sin {\left (2 x \right )}}{32} + \frac {15 x \cos {\left (2 x \right )}}{32} - \frac {15 \sin {\left (2 x \right )}}{64} \]