# 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

The solution is

#### 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.

#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

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

#4 = sqrt(49) + C#

#4 - 7 = C#

#C = -3#

The solution is therefore

Hopefully this helps!