Answer:
[tex]y=2e^{5x}[/tex]
Step-by-step explanation:
The slope of a curve is given by its derivative function. From the question, its value at any point (x, y) is 5 times y.
[tex]\dfrac{dy}{dx}=5y[/tex]
[tex]\dfrac{dy}{y}=5dx[/tex]
Integrate both sides
[tex]\int\dfrac{dy}{y}=5dx[/tex]
Don't forget the constant of integration
[tex]\ln y= 5x + C[/tex]
[tex]y=e^{5x+C}=e^{5x}\cdot e^C[/tex]
Since C is a constant, then [tex]e^C[/tex] is constant. Let's call it A.
[tex]y=Ae^{5x}[/tex]
At the point (0, 2), when x = 0, y = 2.
[tex]2=Ae^{5\times0}[/tex]
[tex]2=Ae^{0}[/tex]
[tex]A=2[/tex]
Hence,
[tex]y=2e^{5x}[/tex]
