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?

Answer 1

Answer:
`y=2x^2`

Explanation:

Given Curve passes through
`\left ( 0,5\right )`

Also the slope of the curve at every point p is twice the Y-coordinate of p

Let the coordinate of p be
`\left ( x,y\right )`

Therefore

`\frac{\mathrm{d} y}{\mathrm{d} x}=2y`

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

Integrating both sides

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

`\ln y=2x+c`

Substituting values

`\ln 5=c`

`\ln (y)/(5)=2x`

`y=2x^2`