Question

Find the particular solution of the differential equation e^x tan y dx + (2 – e^x) sec² y dy = 0, give that `y = pi/4` when x = 0

Answer 1

`e^x tan y dx = (e^x - 2)sec^2y dy`

Using variable seprable

`((sec^2y)/(tan y)) dy = (e^x/(e^x - 2))dx`

Integrating both side

`int ((sec^2 y)/(tany)) dy = int (e^x/(e^x - 2)) dx`

Let tan y=p ⇒ sec²y dy = dp and (e^x−2) = q  ⇒ e^xdx =dq

`int (dp)/p = int (dq)/q`

In (p) = In (q) + In(c),  where c is constant of integration

p = qc

Replacing the values

`tan y = c(x^x - 2)`

When x = 0, `y = pi/4`

1 = c(-1)

c = -1

`tan y = 2 - e^x`

`y = tan^(-1) (2 - e^x)`