23.2.21 problem 6(j)

Internal problem ID [4138]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 3. Linear differential equations of second order. Exercises at page 31
Problem number : 6(j)
Date solved : Sunday, March 30, 2025 at 02:40:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{x} \sin \left (\sqrt {2}\, x \right ) c_2 +{\mathrm e}^{x} \cos \left (\sqrt {2}\, x \right ) c_1 +\frac {x^{3}}{3}+\frac {2 x^{2}}{3}+\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+\frac {2 x}{9}-\frac {8}{27} \]

✓ Mathematica. Time used: 1.388 (sec). Leaf size: 68

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

\[ y(x)\to \frac {x^3}{3}+\frac {2 x^2}{3}+\frac {2 x}{9}+\frac {\sin (x)}{4}+\frac {\cos (x)}{4}+c_2 e^x \cos \left (\sqrt {2} x\right )+c_1 e^x \sin \left (\sqrt {2} x\right )-\frac {8}{27} \]

✓ Sympy. Time used: 0.279 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = \frac {x^{3}}{3} + \frac {2 x^{2}}{3} + \frac {2 x}{9} + \left (C_{1} \sin {\left (\sqrt {2} x \right )} + C_{2} \cos {\left (\sqrt {2} x \right )}\right ) e^{x} + \frac {\sin {\left (x \right )}}{4} + \frac {\cos {\left (x \right )}}{4} - \frac {8}{27} \]