Differential Equations

Question

Find the particular solution of the differential equation:

 

x^2/(y^2-4)*dy/dx = 1/(2y)

 

satisfying the initial condition y(1)=sqrt(5)

y = ?

the answer must be the function of x

Answer 1

[x² / (y² - 4)] dy/dx = 1/(2y)

 

This is a separable equation.

 

[2y / (y² - 4)] dy = dx / x²

 

Integrate both sides.

 

ln(y² - 4) = -1/x + c

 

Take e⁽⁾ of both sides.

 

y² - 4 = e^c e^-1/x = ke^-1/x

 

y(1) = √5 ⇒

 

5 - 4 = k/e

 

1 = k/e

 

k = e

 

y² - 4 = ee^-1/x = e^1-1/x = e^(x-1)/x

 

y² = e^(x-1)/x + 4

 

y = [e^(x-1)/x + 4]^1/2

 

x ≠ 0

 

Answer 2

Separable:

 

   2y dy             dx

-----------  = -----------

 y^2 - 4             x^2

 

 

Integration, left side:

 

Let Z = y^2-4

Then dZ = 2y dy

 

integral dz/z = ln |Z|

 

                    ln| y^2 - 4|

 

Integration right side is -1/X + c

 

Applying the initial condition

 

ln|1| = -1 + c

0 = -1 + c

c= 1

 

So ln|y^2 - 4| = 1 - 1/x

 

solve for y