# First order differential, initial value probelm

## Homework Statement

In problem 11, determine by inspection at least two solutions of the given initial-value problem.

## Homework Equations

11. dy/dx = 3y^(2/3) , y(0)=0

## The Attempt at a Solution

I can't figure out what they mean by inspection, I have no examples to go off from.

If i make it f(x,y)= 3y^(2/3) then the derivative (df/dy) would be 2/(y^(1/3))
then y could not = 0 which means there is not a guaranteed unique solution.

Now how do I go on to get the solutions :/?

Book says y = 0 and y= x^3 is the answer.

Pengwuino
Gold Member
Remember, you're looking for a function y(x) that satisfies the differential equation. In other words, some y(x) that, when differentiated with respect to x once, you get 3 * that function to the 2/3 power of the original function.

Surely given y = 0, dy/dx = 3y^(2/3) right? 0^2/3 = 0, d/dx of 0 is 0 right?

Am I looking for a function when I take the derivative and plug in 0 for x, I get 0?

Not really understanding how they got the y=x^3

dy/dx = 3x^2

Mark44
Mentor
The differential equation, dy/dx = 3y^(2/3), is separable.
dy/dx = 3y^(2/3) ==>
y^(-2/3)dy = 3dx

Now integrate both sides.

That helped thanks.

That method seems to not work on the next problem they gave,

x(dy/dx) = 2y

dy/y = dx/x * 2

ln(y) = 2ln(x)

Book Answer is y = x^2

Is there some other way of solving these ? I am in section 2.1 and separable variables is the name of 2.2. They seem to be explaining how to do #11 in the next section. Any idea what they mean by inspection?

**edit: Forgot about raising e to both sides, that would give me x^2.

Guess this still works :/

Mark44
Mentor
That helped thanks.

That method seems to not work on the next problem they gave,

x(dy/dx) = 2y

dy/y = dx/x * 2

ln(y) = 2ln(x)
You forgot the constant of integration, which leads me to believe you might also have forgotten it in the previous problem. In the other problem, it turned out to be zero, so your error didn't cost you anything.
Book Answer is y = x^2

Is there some other way of solving these ? I am in section 2.1 and separable variables is the name of 2.2. They seem to be explaining how to do #11 in the next section. Any idea what they mean by inspection?

**edit: Forgot about raising e to both sides, that would give me x^2.

Guess this still works :/

vela
Staff Emeritus