Differential Equations Question

Question

Find the particular solution of the differential equation
(x^2/(y^2-6))(dy/dx)=1/(2y) satisfying the initial condition y(1)=√7

 

Could someone help me understand this?

Answer 1

[x²/(y²-6)](dy/dx) = 1/(2y)

 

Separate variables to get [2y/(y²-6)]dy = (1/x²)dx

 

Integrate both sides to obtain:  ln(y²-6) = -1/x + C

 

Since y = √7 when x = 1, we have ln(1) = -1+C.  So, C = 1

 

Therefore, ln(y²-6) = 1-1/x

 

So, y²-6 = e^1-1/x

 

         y² = e^1-1/x +6

 

         y = √(e^1-1/x+6)  (Note:  normally, when you take the square                                       root of both sides, we would need ± in front of                                   the radical.  But, since y(1) = √7, y must be                                       positive.)