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?

1 Answer
May 6, 2017

y=5e^(2x).y=5e2x.

Explanation:

Recall that the slope of the curve at a point P(x,y)" is "dy/dx.P(x,y) is dydx.

Therefore, by what has been given, we have,

dy/dx=2y.dydx=2y.

rArr dy/y=2dx.dyy=2dx.

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

General Solution, we integrate it termwise, and get,

intdy/y=2intdx+lnc,dyy=2dx+lnc,

rArr lny=2x+lnc, or, ln(y/c)=2x.lny=2x+lnc,or,ln(yc)=2x.

rArr y/c=e^(2x).yc=e2x.

:. y=c*e^(2x), is the eqn. of the Curve.

Since, this curve passes through the point (0,5), we must have,

5=c*e^0=c*1 rArr c=5.

Hence, the eqn of the curve is, y=5e^(2x).

Enjoy Maths.!