Solving dy/dx=y+cos(x)y^2010 Using Var. of Parameters/Constant Method

[dooogle]

Homework Statement

solve

dy/dx=y+cos(x)y^2010

using variation of parameters/constant method

Homework Equations

The Attempt at a Solution

let z=1/(y^(n-1))=1/(y^(2009))

dz/dx=(dz/dy)(dy/dx)

dz/dy=-(n-1)*y^(-n)*dy/dx=-(2009*y^-2010)*(y+cos(x)y^2010)

=-2009(y^(-2009)+cos(x))

since z=y^(-2009)

=-2009(z+cos(x))

so dz/dx=-2009z-2009cos(x)

how do i separate the f(z) and f(x) to integrate?

also i need to do the equation by integrating factor but do not know what to take as the integrating factor since normally i rearange an equation into the form dy/dx+p(x)y=q(x)
and take e^int(p(x))dx as the integrating factor

thanks for your time

dooogle

 

[fzero]

The equation is not separable. Rearrange the terms to write

$$\frac{ dz}{dx} + 2009z =-2009cos(x).$$

This is of the form you are used to, just determine p and q.

 

[dooogle]

hi thanks for the help

so i can take p(x)=2009 and q(x)=-2009cos(x)

using e^int(2009)dx=e^2009x

multiply throughout by e^2009x giving

d(e^2009x)z/dx= -2009cos(x)*e^2009x

so e^2009x*z= int -2009cos(x)*e^2009x dx

using the integrating factor method

but what i don't understand is why i need to use both the variation of parameters/constants and the integrating factor methods when the question asks for me to solve the equation using one method then the other and compare the methods

thanks for your help

dooogle

 

[fzero]

Variation of parameters is not what you did. All you've done is a change of variables at this point. The method of variation of parameters is the following. First, find the solution to the homogenous equation

$$\frac{ dz}{dx} + 2009z =0,$$

and call it $$z_h(x)$$. Then look for a particular solution to the inhomogenous equation of the form

$$z_p(x) = c(x) z_h(x) .$$

Usually $$c(x)$$ will satisfy a simpler equation than the one we started with. Then the general solution is

$$z = z_h + z_p.$$

This last expression is what you want to compare to the result of using the integrating factor.
The particular solution $$z_p$$

 

[korobeiniki]

Hi guys, sorry to reopen an absolutely archaic thread but I'm trying to solve an equation similar to this one (well I already have). I'm stuck on the comparison between the integrating factor and variation of constants techniques. As far as I'm aware there is absolutely no difference for a first-order linear equation? Thanks for any help.

 

[LCKurtz]

Hi guys, sorry to reopen an absolutely archaic thread but I'm trying to solve an equation similar to this one (well I already have). I'm stuck on the comparison between the integrating factor and variation of constants techniques. As far as I'm aware there is absolutely no difference for a first-order linear equation? Thanks for any help.

You can't use the variation of parameters method to find a particular solution of the NH equation unless you already have a solution of the homogeneous equation. You have to get that somehow first.

The advantage of the integrating factor method is it gives the general solution (at least in principle) immediately.