20.9.11 problem Problem 11

Internal problem ID [3755]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 11
Date solved : Sunday, March 30, 2025 at 02:07:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\csc \left (x \right )+2 x^{2}+5 x +1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+y(x) = csc(x)+2*x^2+5*x+1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+y[x]==Csc[x]+2*x^2+5*x+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 x^2+5 x+(-x+c_1) \cos (x)+\sin (x) (\log (\sin (x))+c_2)-3 \]

✓ Sympy. Time used: 2.898 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 - 5*x + y(x) + Derivative(y(x), (x, 2)) - 1 - 1/sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - x\right ) \cos {\left (x \right )} + \left (2 x^{2} + 5 x - 3\right ) \cos ^{2}{\left (x \right )} + \left (C_{2} + 2 x^{2} \sin {\left (x \right )} + 5 x \sin {\left (x \right )} + 4 x \cos {\left (x \right )} + \left (- 4 x - 5\right ) \cos {\left (x \right )} + \log {\left (\sin {\left (x \right )} \right )} - 3 \sin {\left (x \right )} + 5 \cos {\left (x \right )}\right ) \sin {\left (x \right )} \]