Find an equation of the curve that satisfies dy/dx=150yx^14 and whose y-intercept is 6.
My work so far:
dy/dx = 50yx^14
ln(y) = 10yx^15+C
y(0)=6
y=10(6)x^15+C
Help?
To find an equation of the curve that satisfies the given conditions, we can use the technique of "separation of variables".
Step 1: Start with the given differential equation: dy/dx = 150yx^14.
Step 2: Rearrange the equation by dividing both sides by y and multiplying both sides by dx: dy/y = 150x^14 dx.
Step 3: Integrate both sides of the equation with respect to their variables. ∫(dy/y) = ∫(150x^14 dx).
The integral of (dy/y) is ln|y|, and the integral of (150x^14 dx) is (150/15)x^15 + C, where C is the constant of integration.
Step 4: Applying the integrals, we get ln|y| = 10x^15 + C.
Step 5: Using the given y-intercept is 6, we substitute the values of x and y into the equation to find the value of C.
When x = 0 and y = 6:
ln|6| = 10(0)^15 + C
ln|6| = C.
Therefore, C = ln|6|.
Step 6: Substitute the value of C into the equation found in Step 4: ln|y| = 10x^15 + ln|6|.
Step 7: Exponentiate both sides of the equation to eliminate the natural logarithm. Recall that e^ln|y| = |y|.
|y| = e^(10x^15 + ln|6|).
Step 8: Simplify the equation by removing the absolute value symbol.
y = ± e^(10x^15 + ln|6|).
Therefore, an equation of the curve that satisfies the given conditions is y = ± e^(10x^15 + ln|6|).
To find an equation of the curve that satisfies the given differential equation and has a y-intercept of 6, we can solve the differential equation and then substitute the y-intercept into the solution.
First, let's solve the differential equation:
Separate the variables and integrate both sides:
dy/dx = 150yx^14
dy/y = 150x^14 dx
Integrate both sides:
∫(1/y) dy = ∫(150x^14) dx
ln|y| = 150 * (1/15)x^15 + C
where C is the constant of integration.
Now, exponentiate both sides:
|y| = e^(150/15 * x^15 + C)
Since the y-intercept is 6, we can substitute the values (x = 0, y = 6) into the equation:
|6| = e^(150/15 * 0^15 + C)
6 = e^C
Thus, e^C = 6, and taking the logarithm of both sides:
C = ln(6)
Therefore, the equation of the curve is:
y = ± e^(150/15 * x^15 + ln(6))
Simplifying further:
y = ± 6 * e^(10x^15)
dy/dx=150yx^14
divide both sides by y
(dy/dx) / y = 150x^14
ln(y) = 10 x^15 + c
but (0,6) lies on it, so
ln 6 = 10(0) + c
c = ln 6
ln y = 10x^15 + ln 6
is one such equation