Integrating Factors for Differential Equations

Nicollkeee

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Mar 19, 2015
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I have to use an integrating factor to obtain the general solution for the differential equation dy/dx + y/x = sinx and the particular solution given y =1 when x = pi

I think I've found the integrating factor to be x leaving x*dy.dx + y/x *x = sinx*x
simplified to x*dy/dx + y = xsin(x)

I'm not sure where to go from here, I know I need to integrate both sides but how do I integrate x*(dy/dx) + y ?
 
I have to use an integrating factor to obtain the general solution for the differential equation dy/dx + y/x = sinx and the particular solution given y =1 when x = pi

I think I've found the integrating factor to be x leaving x*dy.dx + y/x *x = sinx*x
simplified to x*dy/dx + y = xsin(x)

I'm not sure where to go from here, I know I need to integrate both sides but how do I integrate x*(dy/dx) + y ?
What is the derivative of x*y? Use the chain rule.
 
I'm not sure how to do that with y in it? would it just be x + y ?
The chain rule says that if you have the product of two functions (of x), then the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second, that is
\(\displaystyle \dfrac{d(f(x) g(x))}{dx} = \dfrac{df}{dx}\,\, g + f\,\, \dfrac{dg}{dx}\)
Now apply that to the product x y(x).
 
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