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# Solving Differential Equation by Substitution watch

1. The equation is:

dy/dx = 2xy - 4xy^2

I need to solve using the substitution y=1/z

I found that dy/dx = (-1/z^2)*(dz/dx)

Therefore after the substitution I ended up with:

(-1/z^2)(dz/dz) = 2x(1/z) - 4x(1/z^2)

I was then asked to solve by integrating factor:

I rearranged into standard form:

(1/z^2)(dz/dz) * 2x(1/z) = 4x(1/z^2)

I got the integrating factor to be e^x.

I ended up with:
(e^x)(z) = Integral [(e^x)(4x)(1/z^2)]

How can I integrate the left hand side of the equation if it had z's in it, should i convert back to y at this point?

Thanks!
2. You've got the substitution right, couple of things I can see though:

(Original post by _paul)

I rearranged into standard form:

(1/z^2)(dz/dz) * 2x(1/z) = 4x(1/z^2)

Thanks!
Did you mean dz/dx here?

(Original post by _paul)

I rearranged into standard form:

(1/z^2)(dz/dz) * 2x(1/z) = 4x(1/z^2)

I got the integrating factor to be e^x.

I ended up with:
(e^x)(z) = Integral [(e^x)(4x)(1/z^2)]
After multiplying everything by z^2 on your standard form bit, I get:

dz/dx + 2xz = 4x

Which should give an integrating factor of e^(x^2), I'm not sure where the z has came from in your integral at the end, it should just be the function of x * your integrating factor that you're integrating.

Hope this helps, ask some more if you need a bit more help.
3. (Original post by _paul)
The equation is:

dy/dx = 2xy - 4xy^2

I need to solve using the substitution y=1/z

I found that dy/dx = (-1/z^2)*(dz/dx)

Therefore after the substitution I ended up with:

(-1/z^2)(dz/dz) = 2x(1/z) - 4x(1/z^2)

I was then asked to solve by integrating factor:

I rearranged into standard form:

(1/z^2)(dz/dz) * 2x(1/z) = 4x(1/z^2)

I got the integrating factor to be e^x.

I ended up with:
(e^x)(z) = Integral [(e^x)(4x)(1/z^2)]

How can I integrate the left hand side of the equation if it had z's in it, should i convert back to y at this point?

Thanks!
I assume that the first bolded is supposed to be an x.

The second bolded isn't in standard form. It is also incorrect.

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