Question

Solve the following differential equation:

`sin  ("d"y)/("d"x)` = a, y(0) = 1

Answer 1

`sin  ("d"y)/("d"x)` = a

`sin  ("d"y)/("d"x)` = sin^–1(a)

The equation can be written as

dy = sin^–1(a) dx

Taking integration on both sides, we get

`int "d"y = int sin^-1 ("a")  "d"x`

y  = `sin^-1 "a" int "d"x`

y = sin^–1(a) x + C  ........(1)

Initial condition:

Since y (0) = 1, we get

y = sin^–1(a) x + C

1 = sin^–1(a) (0) + C

0 + C = 1

C = 1

Equation (1)

⇒ y = sin^–1(a) x + 1

y – 1 = sin^–1(a) x

`(y - 1)/x` = sin^–1(a)

`sin((y - 1)/x)` = a