# 1st Order ODE

Larrytsai

## Homework Statement

y'(x)^2 = y^2 +xy

## Homework Equations

hint : let u = y/x

## The Attempt at a Solution

I divide x^2 on both sides and end up with...

y'= y^2/x^2 + y/x

then using the hint, i get
y' = u^2 + u

but i do not know how to solve the DE from here.

Homework Helper

## The Attempt at a Solution

I divide x^2 on both sides and end up with...

y'= y^2/x^2 + y/x

then using the hint, i get
y' = u^2 + u

but i do not know how to solve the DE from here.

You should have

(1/x2)y'2 =u2+u

if u=y/x or y=ux, what is y' ?

viscousflow
whoops just gave the answer, don't want to destroy your thunder rock.

Larrytsai
You should have

(1/x2)y'2 =u2+u

if u=y/x or y=ux, what is y' ?

hmm how did u get y'/x^2?

EDIT:

Im a little confused, when taking the derivative of y= ux, am i differentiating w.r.t x?

Homework Helper
hmm how did u get y'/x^2?

EDIT:

Im a little confused, when taking the derivative of y= ux, am i differentiating w.r.t x?

yes, you differentiate with respect to x.

The left side was y'2, so dividing both sides by x2 means that the left side becomes y'2/x2

Larrytsai
yes, you differentiate with respect to x.

The left side was y'2, so dividing both sides by x2 means that the left side becomes y'2/x2

ohh sorry, only the x was squared for the left term not the y'

Larrytsai
k so taking the derivative of y=ux, i have y' = xu' + u, and i just sub it into the equation, then i have xu' = u^2 + u which I can simply put it in the general first oder ODE form, and solve by separation

Homework Helper
ohh sorry, only the x was squared for the left term not the y'

k so taking the derivative of y=ux, i have y' = xu' + u, and i just sub it into the equation, then i have xu' = u^2 + u which I can simply put it in the general first oder ODE form, and solve by separation

I have small query, is the term on the left yx2 OR (y')2 ?

Because y(x) is a function in x and y'(x) is its derivative, so y'(x)2 is the derivative squared, so which is it?

Larrytsai
I have small query, is the term on the left yx2 OR (y')2 ?

Because y(x) is a function in x and y'(x) is its derivative, so y'(x)2 is the derivative squared, so which is it?

ohh yah, I meant y' * x^2

Homework Helper
ohh yah, I meant y' * x^2

ah in that case, this post under this one is correct.

k so taking the derivative of y=ux, i have y' = xu' + u, and i just sub it into the equation, then i have xu' = u^2 + u which I can simply put it in the general first oder ODE form, and solve by separation

You will get xu' + u = u2+u, the 'u' will cancel and then you can solve by separation. then you need to remember that you will get 'u' and not 'y', but you know that y=ux.

Larrytsai
ah in that case, this post under this one is correct.

You will get xu' + u = u2+u, the 'u' will cancel and then you can solve by separation. then you need to remember that you will get 'u' and not 'y', but you know that y=ux.

great! thanks a lot for your help