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:$y=2x^2$

Step-by-step 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

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

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

Integrating both sides

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

$\ln y=2x+c$

Substituting values

$\ln 5=c$

$\ln \frac{y}{5}=2x$

$y=2x^2$