Find the Function​ y(x) Satisfying

Question

Find the function​ y(x) satisfying dy/dx=5x^−8/9 and ​y(−​1)=−6.

y(x)= ?

Answer 1

For this problem, we separate each part of the first-order derivative and then integrate each side as follows:

∫dy = ∫5x^-8/9dx.

Integrate both sides, and let C = C₁ + C₂ (constants are arbitrary in this case):

y + C₁ = 45x^1/9 + C₂ => y = 45x^1/9 + C.

Then, set the equation equal to -6 and substitute x=-1:

y(-1) = -6 = 45(-1)^1/9+C

Simplify for C and find it's value:

C = 39.

Thus, the general function y(x) = 45x^1/9 + 39.

Answer 2

This type of question is sometimes called an initial value problem since they've given you a value for y(x), specifically y(-1) = -6. They also supplied us with a differential equation dy/dx = 5x^-8/9.

To solve, we'll start by "separating" the differential equation and just multiplying the dx to the other side of the equation: dy = 5x^-8/9dx

Now we find the antiderivative of both sides.

∫dy = ∫5x^-8/9dx

The left side just gets us y, while the right side(applying the power rule) we should end up with 45x^1/9 + C.

If you have questions about using the power rule, you can ask me further later, but I'm operating under the assumption you are familiar.

so y = 45x^1/9 + C

Don't forget the C! Because now we need to find the particular C that works with out initial condition. So substitute the information y(-1) = -6 and solve for C.

-6 = 45(-1)^1/9 + C

-6 = -45 + C

39 = C

So finally our complete solution should be y = 45x^1/9 + 39

Answer 3

To take a derivative using the power rule, you subtract 1 from the exponent. To go backwards, you would add 1 to the exponent. Adding 1 to -8/9 would give you 1/9 so the original function must be a constant C multiplied by x^1/9 or Cx^1/9

Taking the derivative of this (using the power rule) would give us (1/9)(C)x^-8/9 but we know that the coefficient in front of the x^-8/9 is actually 5 so (1/9)(C) = 5 or C = 45. That makes the original function y(x) = 45x^1/9 + C

Since we are told y(−​1) = −6, then:

-6 = 45(-1)^1/9 + C

-6 = 45(-1) + C

-6 = -45 + C

C = 39

So y(x) = 45x^1/9 + 39

Answer 4

5x^(-8/9)

integrate to get

5x^(1/9)/(1/9) + c

= 45x(1/9) + c

= 45(-1)^(1/9) +c

= -45 +c

=-6

c = -6 +45 = 39

c= 39