Calculus, Differential Equations

Question

Solve the following differential equation for y=f(1)=3

dy/dx=2/xy

Answer 1

dy/dx = 2/xy

cross multiplies:

x y dy = 2 dx

y dy = 2/x dx

integrates:

(1/2)y^2 = 2 ln x + c

y^2 = 4 ln x + 2c

y = sqrt(4 ln x + 2c)

(1,3) --> 3 = sqrt( 4 ln 1 + 2c)

3 = sqrt( 4 * 0 + 2c)

3 = sqrt(2c)

9 = 2c

c = 9/2

solution:

y = sqrt(4 ln x + 9)

check:

dy/dx = (1/2)(4 ln x + 9)^(-1/2) ( 4/x)

dy/dx = (2/x)(4 ln x + 9)^(-1/2)

= 2/ (x *sqrt( 4 lnx + 9))

is the left side of the original equation

right side is 2/(xy) = 2/ (x sqrt(4 ln x + 9))

so the solution checks.

solution:

y = sqrt(4 ln x + 9)