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प्रश्न
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) = x + y`
उत्तर
`x ("d"y)/("d"x) = x + y`
⇒ `("d"y)/("d"x) = (x + y)/x` .......(1)
It is a homogeneous differential equation, Same degree in x and y
Put y = vx and `("d"y)/("d"x) = "v" + x ("d"y)/("d"x)`
Equation (1)
⇒ `"v" + x ("d"y)/("d"x) = (x + "v"x)/x = (x(1 + "v"))/x`
`"v" + x "dv"/("d"x) = (1 + "v")`
⇒ `x ("d"y)/("d"x) = 1 + "v" - "v"`
`x "dv"/("d"x)` = 1
dv = `1/x "d"x`
Integrating on both sides
`int 1/x "d"x = int "dv"`
log x = v + c
⇒ x = `"e"^("v" + "c")`
x = ev. ec
x = ev. c
⇒ x = cev ......[⇒ `"v" = y/x]`
⇒ x = `"ce"^(y/x)`
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