`y' = 1/(xsqrt(4x^2-9))` Solve the differential equation.

`y'=1/(xsqrt(4x^2-9))`

`y=int1/(xsqrt(4x^2-9))dx`

Apply integral substitution: `x=3/2sec(u)`

`dx=3/2sec(u)tan(u)du`

`y=int1/(3/2sec(u)sqrt(4(3/2sec(u))^2-9))(3/2sec(u)tan(u))du`

`y=inttan(u)/sqrt(4(9/4sec^2(u))-9)du`

`y=inttan(u)/sqrt(9sec^2(u)-9)du`

`y=inttan(u)/(sqrt(9)sqrt(sec^2(u)-1))du`

Now use the identity:`sec^2(x)=1+tan^2(x)`

`y=inttan(u)/(3sqrt(1+tan^2(u)-1))du`

`y=inttan(u)/(3sqrt(tan^2(u)))du`

`y=inttan(u)/(3tan(u))du` assuming `tan(u) >=0`

`y=int1/3du`

take the constant out,

`y=1/3intdu`

`y=1/3u`

Substitute back `u=arcsec((2x)/3)`

and add a constant C to the solution,

`y=1/3arcsec((2x)/3)+C`

Comments

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