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September 23rd, 2015, 09:21 AM  #1 
Member Joined: Nov 2014 From: SG Posts: 55 Thanks: 1  Does anyone know how to solve this ODE question? 
September 23rd, 2015, 10:14 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
That looks like a fairly straightforward "second order, nonhomogeneous, differential equation with constant coefficients". First find the general solution to the "associated homogeneous equation", $\displaystyle y''+ 4y= 0$. The "characteristic equation" is $\displaystyle r^2+ 4= 0$ which has roots $\displaystyle \pm 2i$. That means that the general solution to the homogeneous equation is $\displaystyle y(x)= C_1 \cos(2x)+ C_2 \sin(2x)$. Now we need to find a single solution to the entire equation. Normally, seeing "cos(2x)" on the right we would try "y(x)= A cos(2x)+ B sin(2x)" but that is already a solution to the homogeneous equation. So we try instead "y(x)= Ax cos(2x)+ Bx sin(2x)". With that y, what is y''? Putting that into the differential equation, what equation for A and B do you get? Last edited by skipjack; September 23rd, 2015 at 10:58 AM. 
September 23rd, 2015, 10:44 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,476 Thanks: 2039 
Would you (KaiL) be able to finish if we suggest how you can start? For example, you could start by multiplying the equation by sin(2x), which gives an equation that can be integrated to give a first order ODE.


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