# 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?

May 6, 2017

$y = 5 {e}^{2 x} .$

#### Explanation:

Recall that the slope of the curve at a point $P \left(x , y\right) \text{ is } \frac{\mathrm{dy}}{\mathrm{dx}} .$

Therefore, by what has been given, we have,

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

$\Rightarrow \frac{\mathrm{dy}}{y} = 2 \mathrm{dx} .$

This is a Separable Variable Type Diff. Eqn., and, to find its

General Solution, we integrate it termwise, and get,

$\int \frac{\mathrm{dy}}{y} = 2 \int \mathrm{dx} + \ln c ,$

$\Rightarrow \ln y = 2 x + \ln c , \mathmr{and} , \ln \left(\frac{y}{c}\right) = 2 x .$

$\Rightarrow \frac{y}{c} = {e}^{2 x} .$

$\therefore y = c \cdot {e}^{2 x} ,$ is the eqn. of the Curve.

Since, this curve passes through the point $\left(0 , 5\right) ,$ we must have,

$5 = c \cdot {e}^{0} = c \cdot 1 \Rightarrow c = 5.$

Hence, the eqn of the curve is, $y = 5 {e}^{2 x} .$

Enjoy Maths.!