Question

A curve passes through the point (0, 5) 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? (Use x as the independent variable.

Answer 1

`y=5e^(2x)`

Explanation:

Let (x,y) represents a point P on the curve,

So, the slope of the curve at point P =
`(dy)/(dx)`

According to the question,

`(dy)/(dx)=2y`

`(1)/(y)dy=2dx`

Integrating both sides,

`\int (dy)/(y)=2dx`

`ln y=2x+ln C`

`ln y-ln C = 2x`

`ln((y)/(C))=2x`

`(y)/(C)=e^(2x)`

`\implies y=Ce^(2x)`

Since, the curve is passing through the point (0, 5),

`5=Ce^(0)\implies C=5`

Hence, the required equation of the curve is,

`y=5e^(2x)`