Equation of a Curve Passing Through (0, 9)

[locachola17]

Homework Statement

A curve passes through the point (0, 9) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

Homework Statement

A curve passes through the point (0, 9) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?

Homework Equations

The Attempt at a Solution

Hi locachola, welcome to PF!

We're not here to do your homework for you, we're here to help you learn. You must show some attempt at a solution, in order to receive help...

 

[lanedance]

Hi locachola17 wlecome to PF

The idea of the forum is to help with your working... so any ideas on how to get started?

as a hint, the line
"has the property that the slope of the curve at every point P is twice the y-coordinate of P"
can you write this line as an equation?

 

[locachola17]

okay well at the point (0,9) the y-coordinate is 9
if the slope is twice the y-coordinate, the slope is 18?!

 

[gabbagabbahey]

okay well at the point (0,9) the y-coordinate is 9
if the slope is twice the y-coordinate, the slope is 18?!

Sure, but what about at any other point (x,y) on the curve? What does that tell you about dy/dx for this curve?

 

[lanedance]

thats for that single point...

but the question says its true for every point on the line... how would you write this to show it for every point on the line...?

remeber the slope is given by the derivative y'(x) = dy/dx

 

[locachola17]

dy/dx is 2y

 

[gabbagabbahey]

dy/dx is 2y

Right, so y(x)=____?

 

[gabbagabbahey]

but the question says its true for every point on the line...

No it doesn't...It says for every point on the "curve"...the curve is not a line

 

[locachola17]

y(x)=y^2

 

[gabbagabbahey]

y(x)=y^2

That makes no sense, you have a separable differential equation for y(x): $$\frac{dy(x)}{dx}=2y(x)$$...how do you usually solve a separable DE?

 

[locachola17]

you integrate both sides?

 

[gabbagabbahey]

you integrate both sides?

How do you determine $$\int y(x)dx$$ when you don't know what y(x) is?

No, you separate varaiables first and then integrate:

$$\frac{dy}{dx}=2y\implies \frac{dy}{y}=2dx$$

Now you can integrate...