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?

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Answer 1 · 2017-05-06T17:04:02

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

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

Therefore, by what has been given, we have,

#dy/dx=2y.#

# rArr dy/y=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,#

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

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

#:. 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.!