15.15.11 problem 12

Internal problem ID [3215]
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 : 12
Date solved : Sunday, March 30, 2025 at 01:21:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=x^{2} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+3*y(x) = x^2*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} c_2 +{\mathrm e}^{x} c_1 +\frac {\left (25 x^{2}+55 x +28\right ) \cos \left (x \right )}{125}+\frac {\sin \left (x \right ) \left (25 x^{2}-20 x -67\right )}{250} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 52

ode=D[y[x],{x,2}]-4*D[y[x],x]+3*y[x]==x^2*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{250} \left (\left (25 x^2-20 x-67\right ) \sin (x)+2 \left (25 x^2+55 x+28\right ) \cos (x)\right )+c_1 e^x+c_2 e^{3 x} \]

✓ Sympy. Time used: 0.307 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*sin(x) + 3*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x} + C_{2} e^{3 x} + \frac {x^{2} \sin {\left (x \right )}}{10} + \frac {x^{2} \cos {\left (x \right )}}{5} - \frac {2 x \sin {\left (x \right )}}{25} + \frac {11 x \cos {\left (x \right )}}{25} - \frac {67 \sin {\left (x \right )}}{250} + \frac {28 \cos {\left (x \right )}}{125} \]