76.15.34 problem 35

Internal problem ID [17596]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 35
Date solved : Thursday, March 13, 2025 at 10:37:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

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

\[ y = \sin \left (2 \ln \left (x \right )\right ) c_{2} +\cos \left (2 \ln \left (x \right )\right ) c_{1} +\frac {\sin \left (\ln \left (x \right )\right )}{3} \]

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 29

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

\[ y(x)\to \frac {1}{3} \sin (\log (x))+c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x)) \]

✓ Sympy. Time used: 0.326 (sec). Leaf size: 26

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

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