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
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!