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

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

Recall that the slope of the curve at a point $P (x, y) is \frac{d y}{d x} .$ $P(x,y)" is "dy/dx.$

Therefore, by what has been given, we have,

$\frac{d y}{d x} = 2 y .$ $dy/dx=2y.$

$\Rightarrow \frac{d y}{y} = 2 d x .$ $rArr dy/y=2dx.$

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

General Solution, we integrate it termwise, and get,

$\int \frac{d y}{y} = 2 \int d x + ln c,$ $intdy/y=2intdx+lnc,$

$\Rightarrow ln y = 2 x + ln c, or, ln (\frac{y}{c}) = 2 x .$ $rArr lny=2x+lnc, or, ln(y/c)=2x.$

$\Rightarrow \frac{y}{c} = e^{2 x} .$ $rArr y/c=e^(2x).$

$:. y=c*e^(2x),$ 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.!