Question

A curve passes through the point \(\left( {0,5} \right)\) 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?

Short answer

The required equation of the curve is \(y = 5{e^{2x}}\).

Step-by-step solution

Exponential Growth and Decay

According to the theorem, the only solutions of the differential equation \(\frac{{dy}}{{dt}} = ky\) are the exponential functions \(y\left( t \right) = y\left( 0 \right){e^{kt}}\).

According to a given condition, \(\frac{{dy}}{{dx}} = 2y\) then by theorem \(y\left( x \right) = C{e^{2x}}\).

Determine the equation of the curve

Substitute the point into the equation \(y\left( x \right) = C{e^{2x}}\) and simplify.

\(\begin{aligned}5 &= C{e^{2\left( 0 \right)}}\\5 &= C\left( 1 \right)\\C &= 5\end{aligned}\)

Substitute the value of \(y\left( x \right) = C{e^{2x}}\) into the \(y\left( x \right) = C{e^{2x}}\) to find the equation of the curve.

\(\begin{aligned}y\left( x \right) &= 5{e^{2x}}\\y &= 5{e^{2x}}\end{aligned}\)

Thus, the required equation of the curve is \(y = 5{e^{2x}}\).