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प्रश्न
Solve the following differential equation:
`sin ("d"y)/("d"x)` = a, y(0) = 1
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उत्तर
`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
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