Differential Equations

Question

Solve the separable differential equation 

7x - 6y*sqrt(x^2+1)dy/dx = 0

 

Subject to the initial condition: y(0)=5

y = ?

Answer 1

7x - 6*y*Sqrt(x^2 +1) dy/dx = 0

7x dx - 6 * y * sqrt(x^2 +1 ) dy = 0


7x dx = 6 * y * sqrt(x^2 +1) dy

7x dx 6 y dy
------------ =
sqrt(x^2 +1)


Integration, Left side:

Let Z = x^2 + 1

dZ = 2x dx

1/2 dZ = x dx

7 integral 1/2 dZ / sqrt(z) =

7/2 integral z^(-1/2) dZ =

7/2 integral 2 * z^(1/2)

7 integral z^(1/2)

7 integral (x^2 + 1)^(1/2)

Integration, right side:

3 y ^2 + C


Applying the initial condition:

1 = 96 + c

c = -95