Question

Solve the following homogeneous differential equation:

`x ("d"y)/("d"x) = x + y`

Answer 1

`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 = e^v. e^c

x = e^v. c

⇒ x = ce^v   ......[⇒ `"v" = y/x]`

⇒ x = `"ce"^(y/x)`