Solve: y' = y - xy^3e^-2x?

Question

hehehe.. cnxa n po badly nided.. tnx so much s mga sasagot! ^_^

Answer 1

First-order equations of the form dy/dx+P(x)y=Q(x)y^n are called Bernoulli equations. If n=0 or 1, then the equation is linear. If n is not 0 or 1, the equation is nonlinear but can be reduced to a linear equation with the subsitution w=y^(1-n).

For this problem, n=3 so we let w=y^(-2) -> dw/dx=-2y^(-3) dy/dx -> dy/dx=-1/ 2y^3 dw/dx.
Note that because w=y^(-2), y^3w=y. With this subsitution, the equation

dy/dx=y-x e^(-2x) y^3

becomes

-1/ 2y^3 dw/dx=y^3w-x e^(-2x) y^3

and dividing through by y^3 gives us

-1/2 dw/dx=w- x e^(-2x)

which in standard form is

dw/dx+2w=2 x e^(-2x)

For this linear equation, the integrating factor is e^(2x) and multiplying through by the integrating factor gives us

e^(2x) dw/dx+2 e^(2x) w=2x

so

d/dx (e^(2x) w)=2x

Integrating this gives us

e^(2x) w=x^2+C

and multiplying through by e^(-2x) gives us w:

w=x^2 e^(-2x)+Ce^(-2x)

Since w=1/y^2,

1/y^2=x^2 e^(-2x)+Ce^(-2x)

so

y^2=1/(e^(-2x) (x^2+c))

so

y= \pm e^x/\sqrt{x^2+C}

Remember: \sqrt{e^(-2x)}=\sqrt{(e^(-x))^2}=e^(-x); 1/e^(-x)=e^x.

To make sure that you understand, you should try to solve another one on your own! :)

Answer 2

You subtract 2x from each and each part, offering you with this -y=2-2x then you definitely are making each thing damaging with the intention to get the y sensible, offering you with y=-2+2x it truly is to this point as you may bypass because you won't be able to remedy for 2 variables interior of a similar equation.

Answer 3

Looks like some weird form of Greens Function.


Doug

Answer 4

what?... can't understand the rest after the first part...

And is that a diff eq? Do you want the particular and homogeneous solution...

I got an idea... do your own homework!

Answer 5

I am thinking you are talking about solving differential equation.
learned that long time ago! cant remember from back of my head!

Answer 6

what, u want me to integrate it? dont understand the questions, give some directions.

Answer 7

y=y-2x/3e

Answer 8

cant understand ur ques..

Answer 9

what do you mean ?

Answer 10

Nope.STUDY and YOU answer it!