81.11.21 problem 15-20

Internal problem ID [21645]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 15. Method of undetermined coefficients. Page 337.
Problem number : 15-20
Date solved : Thursday, October 02, 2025 at 07:59:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+8 y&=\left (10 x^{2}+21 x +9\right ) \sin \left (3 x \right )+x \cos \left (3 x \right ) \end{align*}

✓ Maple. Time used: 3.267 (sec). Leaf size: 297

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+8*y(x) = (10*x^2+21*x+9)*sin(3*x)+x*cos(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1000 \left (x^{4}+\frac {21}{5} x^{3}+\frac {311}{50} x^{2}+\frac {189}{50} x +\frac {81}{100}\right ) \left (\left (-3+i\right ) x^{4}+\left (4-14 i\right ) x^{3}+\left (\frac {87}{10}+\frac {389 i}{10}\right ) x^{2}+\left (-\frac {234}{5}-\frac {188 i}{5}\right ) x +\frac {239}{5}+\frac {123 i}{5}\right ) {\mathrm e}^{i \left (-\arctan \left (10 x +21\right )+\arctan \left (\frac {140}{3} x^{2}+\frac {532}{9} x +21+\frac {100}{9} x^{3}\right )+3 x \right )}-1000 \left (\left (3+i\right ) x^{4}+\left (\frac {62}{5}+\frac {24 i}{5}\right ) x^{3}+\left (\frac {909}{50}+\frac {373 i}{50}\right ) x^{2}+\left (\frac {279}{25}+\frac {108 i}{25}\right ) x +\frac {243}{100}+\frac {81 i}{100}\right ) \left (x^{4}-\frac {13}{5} x^{3}+\frac {17}{2} x^{2}-\frac {93}{5} x +17\right ) {\mathrm e}^{i \left (\arctan \left (10 x +21\right )-\arctan \left (\frac {140}{3} x^{2}+\frac {532}{9} x +21+\frac {100}{9} x^{3}\right )-3 x \right )}+400 \left (\frac {x^{4}}{2}+\left (\frac {2}{5}+i\right ) x^{3}+\left (-\frac {21}{100}+\frac {3 i}{100}\right ) x^{2}+\left (\frac {257}{200}-\frac {611 i}{200}\right ) x +\frac {18}{25}-\frac {171 i}{100}\right ) \sqrt {100 x^{4}+420 x^{3}+622 x^{2}+378 x +81}\, \left (\cos \left (\frac {\sqrt {31}\, x}{2}\right ) c_1 +\sin \left (\frac {\sqrt {31}\, x}{2}\right ) c_2 \right ) {\mathrm e}^{-\frac {x}{2}}}{\sqrt {100 x^{4}+420 x^{3}+622 x^{2}+378 x +81}\, \left (200 x^{4}+\left (160+400 i\right ) x^{3}+\left (-84+12 i\right ) x^{2}+\left (514-1222 i\right ) x +288-684 i\right )} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 85

ode=D[y[x],{x,2}]+D[y[x],x]+8*y[x]==(10*x^2+21*x+9)*Sin[3*x]+x*Cos[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 (-\sin (3 x))+\left (-3 x^2+2 x-1\right ) \cos (3 x)+7 x \sin (3 x)-13 \sin (3 x)+c_2 e^{-x/2} \cos \left (\frac {\sqrt {31} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {31} x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.264 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*cos(3*x) - (10*x**2 + 21*x + 9)*sin(3*x) + 8*y(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x^{2} \sin {\left (3 x \right )} - 3 x^{2} \cos {\left (3 x \right )} + 7 x \sin {\left (3 x \right )} + 2 x \cos {\left (3 x \right )} + \left (C_{1} \sin {\left (\frac {\sqrt {31} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {31} x}{2} \right )}\right ) e^{- \frac {x}{2}} - 13 \sin {\left (3 x \right )} - \cos {\left (3 x \right )} \]