29.6.21 problem 21

Internal problem ID [7306]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 6. SECOND-ORDER LINEAR EQUATIONSWITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO. page 422
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 04:28:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 5 y^{\prime \prime }+6 y^{\prime }+2 y&=x^{2}+6 x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=5*diff(diff(y(x),x),x)+6*diff(y(x),x)+2*y(x) = x^2+6*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {3 x}{5}} \sin \left (\frac {x}{5}\right ) c_2 +{\mathrm e}^{-\frac {3 x}{5}} \cos \left (\frac {x}{5}\right ) c_1 +\frac {x^{2}}{2}-\frac {5}{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 47

ode=5*D[y[x],{x,2}]+6*D[y[x],x]+2*y[x]==x^2+6*x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (x^2-5\right )+c_2 e^{-3 x/5} \cos \left (\frac {x}{5}\right )+c_1 e^{-3 x/5} \sin \left (\frac {x}{5}\right ) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - 6*x + 2*y(x) + 6*Derivative(y(x), x) + 5*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{2} + \left (C_{1} \sin {\left (\frac {x}{5} \right )} + C_{2} \cos {\left (\frac {x}{5} \right )}\right ) e^{- \frac {3 x}{5}} - \frac {5}{2} \]