How do you use the differential equation dy/dx=(2x)/sqrt(2x^2-1) to find the equation of the function given point (5,4)?

1 Answer
Mar 4, 2017

The solution is y = sqrt(2x^2 - 1) - 3.

Explanation:

This is a separable differential equation.

dy = (2x)/sqrt(2x^2 - 1) dx

Integrate both sides.

int dy = int (2x)/sqrt(2x^2 - 1) dx

It's true that trig substitution could be used to solve this integral, but a substitution would be easier.

:. Let u = 2x^2 - 1. Then du = 4x dx and dx = (du)/(4x).

int dy = int (2x)/sqrt(u) * (du)/(4x)

int dy = 1/2int 1/sqrt(u)

int dy = 1/2int u^(-1/2)

y = 1/2(2u^(1/2)) + C

y = u^(1/2) + C

y = (2x^2 - 1)^(1/2) + C

We now solve for C using the information given. We know that when x = 5, y = 4, so we can say the following:

4 = sqrt(2(5)^2 - 1) + C

4 = sqrt(49) + C

4 - 7 = C

C = -3

The solution is therefore y = sqrt(2x^2 - 1) - 3.

Hopefully this helps!