Please help solve ODE questions for DE newbie

1. y^{2}(dx/dy) + xy -4y^{2 }=1

I try to simplify to

(dx/dy) + x/y -4 = 1/y^{2 }(dy/dx)^{ = }y^{2} - x/y + 1/4

I don't know how to go next (I try to use integrating factor but it's not working)

2. dy/dx = y(y+3)

(dy/dx)/y(y+3) = 1

Integral (dy/dx)/y(y+3) = Integral 1 dx

then I get

1/3 In(y) - 1/3 In(y+3) = x + c

(I'm not sure about this) My calculus is rusty. I don't remember how to get this number I just remember the formula

Someone please explain or correct. :(

Then I'm not sure how to get

y = .......

Re: Please help solve ODE questions for DE newbie

Quote:

Originally Posted by

**angelme** 1. y^{2}(dx/dy) + xy -4y^{2 }=1

I try to simplify to

(dx/dy) + x/y -4 = 1/y^{2 }(dy/dx)^{ = }y^{2} - x/y + 1/4

I don't know how to go next (I try to use integrating factor but it's not working)

2. dy/dx = y(y+3)

(dy/dx)/y(y+3) = 1

Integral (dy/dx)/y(y+3) = Integral 1 dx

then I get

1/3 In(y) - 1/3 In(y+3) = x + c

(I'm not sure about this) My calculus is rusty. I don't remember how to get this number I just remember the formula

Someone please explain or correct. :(

Then I'm not sure how to get

y = .......

For 1) the integrating factor should work.

The integrating factor that I get is If you don't get this integrating factor post your work so we can see what went wrong.

For 2)

Multiply by 3 to get

This is quadratic in y. If you really want to isolate y you can use the quadratic formula.

Re: Please help solve ODE questions for DE newbie

For #1, let me use w instead of x (w=x). It hurts my eyes otherwise.

.

(Doing this isn't ideal, since y=0 now becomes a bigger problem. But it's a minor concern.)

Yes, the integrating factor is based on 1/y, so is actually , so use .

With practice, that's "seeable" in the initial problem, . Here it amounts to observing that .

So only divide both sides of by just one power of to get:

, so . Then you're in business.

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For #2, is it the partial fractions that you're unsure of? It works like this:

Find .

Algebra: for some constants and (it's the opposite of finding common demoninators.)

It's then straightforward to find and :

implies (clear demoninators)

, implies for all y, implies .

Thus .

Thus

Thus

(And you could simplify more: )