Question

A curve passes through the point (0,2) and has the property that the slope of the curve at every point PP is five times the yy-coordinate of PP. What is the equation of the curve?

Answer 1

Answer:

$y=2e^{5x}$

Step-by-step explanation:

The slope of a curve is given by its derivative function. From the question, its value at any point (x, y) is 5 times y.

$\dfrac{dy}{dx}=5y$

$\dfrac{dy}{y}=5dx$

Integrate both sides

$\int\dfrac{dy}{y}=5dx$

Don't forget the constant of integration

$\ln y= 5x + C$

$y=e^{5x+C}=e^{5x}\cdot e^C$

Since C is a constant, then $e^C$ is constant. Let's call it A.

$y=Ae^{5x}$

At the point (0, 2), when x = 0, y = 2.

$2=Ae^{5\times0}$

$2=Ae^{0}$

$A=2$

Hence,

$y=2e^{5x}$