A curve passes through the point (0, 2) and has the property that
the slope of the curve at every point P is three times the y-coordinate
of P. Find an equation of the curve.
dy/dp = 3y
or ∫ (3/y) dy = ∫ dp
or 3 ln(y) = p + c
or @ (0,2)
ln(2) = 0 + c
or c = ln(2)
3 ln(y) = p + ln(2)
or y = e^( (p + ln(2)) ^(1/3) )
Odd notation. P is a point (x,y) so usually we would see
dy/dx = 3y
y = c*e^3x
2 = c, so
y = 2e^(3x)
You divided the right side by 3 and multiplied the left aside by 3, but that was not correct:
dy/dp = 3y
∫dy/y = ∫3dp
...
To find the equation of the curve, we can start by using the given property of the slope at every point P. The slope of the curve at a point P is equal to three times the y-coordinate of that point.
Let's denote the slope as dy/dp, where p is the independent variable. Since the slope is three times the y-coordinate, we can write:
dy/dp = 3y
Now, we can separate the variables by multiplying both sides by dp and dividing both sides by y:
dy/y = 3dp
Next, we integrate both sides of the equation. On the left side, we integrate with respect to y, and on the right side, we integrate with respect to p:
∫ (1/y) dy = ∫ 3 dp
Integrating the left side gives us:
ln|y| = 3p + C
Here, C is the constant of integration.
To eliminate the absolute value, we can consider the case when y is positive. In that case, we can drop the absolute value:
ln(y) = 3p + C
Now, let's use the given point (0, 2) to find the value of the constant C. When we substitute p = 0 and y = 2 into the equation, we get:
ln(2) = 0 + C
C = ln(2)
Now, we can substitute the value of C back into the equation:
ln(y) = 3p + ln(2)
Next, we can exponentiate both sides of the equation using the base e:
e^(ln(y)) = e^(3p + ln(2))
y = e^(3p) * e^(ln(2))
Using the properties of exponents, we can simplify further:
y = 2 * e^(3p)
So, the equation of the curve that passes through the point (0, 2) and has the property that the slope of the curve at every point P is three times the y-coordinate of P is given by:
y = 2 * e^(3p)