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Question
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Solution
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
∴ `("d"y)/("d"x) = 1 + y/x - (y/x)^2` .....(i)
Put `y/x` = t .....(ii)
∴ y = tx
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "t" + x ("dt")/("d"x)` .....(iii)
Substituting (ii) and (iii) in (i), we get
`"t" + x "dt"/("d"x)` = 1 + t − t2
∴ `x "dt"/("d"x)` = 1 − t2
∴ `"dt"/(1 - "t"^2) = ("d"x)/x`
Integrating on both sides, we get
`int "dt"/(1 - "t"^2) = int ("d"x)/x`
∴ `1/2 log|(1 + t)/(1 - t)|` = log |x| + log |c1|
∴ `log |(1 + y/x)/(1 - y/x)|` = 2log |x| + 2log |c1|
∴ `log|(x + y)/(x - y)|` = log |x2| + log |c12|
∴ `log|(x + y)/(x - y)|` = log |c1x2|
∴ `(x + y)/(x - y)` = c12x2
∴ `(x + y)/(x - y)` = cx2, where c = c12
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