A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$. What is the equation of the curve?
A) $y = 2 e ^ { 3 x + 24 }$
B) $y = \frac { e ^ { 3 x - 24 } } { 2 }$
C) $y = 2 e ^ { 3 x - 8 }$
D) $y = 2 e ^ { 3 x - 24 }$
E) $y = 2 e ^ { x - 24 }$

Question

Quizplus · Accepted Answer

We can start by applying the derivative formula, which for this curve gives us: 
$$\frac{dy}{dx}=3y$$
This is a separable differential equation, and we can solve it by separating the variables, and integrating both sides:
$$\frac{1}{y} dy = 3 dx$$
$$\ln{y} = 3x + C$$
where C is the constant of integration. 
To find C, we can use the fact that the curve passes through (8,2). Plugging these values into the equation above, we get:
$$\ln{2} = 3\cdot 8 + C \Rightarrow C = \ln{2}-24$$
Substituting this value of C back into our equation, we get:
$$\ln{y} = 3x + \ln{2} - 24$$
$$\ln{y} - 3x = \ln{\left(\frac{2}{e^{24}}ight)}$$
$$\ln{y} - 3x = \ln{\left(\frac{2}{\operatorname{exp}(24)}ight)}$$
$$\ln{y} - 3x = \ln{\left(\frac{1}{24\operatorname{log}(e)}\cdot 2ight)}$$
$$\ln{y} - 3x = \ln{\left(\frac{2}{24}ight)}$$
$$\ln{y} - 3x = \ln{\left(\frac{1}{12}ight)}$$
$$\ln{\left(\frac{y}{12}ight)} = 3x$$
$$\frac{y}{12} = e^{3x}$$
$$y = 12e^{3x}$$
Thus, the equation of the curve is $y=12e^{3x}$, which corresponds to choice $	extbf{(D)}$.

A Curve Passes Through the Point (8,2)( 8,2 )(8,2) And Has the Property That the Slope of the Curve

A Curve Passes Through the Point $( 8,2 )$ And Has the Property That the Slope of the Curve