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Question
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
Solution
Given equation is `x^2 "dy"/"dx"` = x2 + xy + y2
⇒ `"dy"/"dx" = (x^2 + xy + y^2)/x^2`
Put y = vx ......[∵ it is a homogeneous differential equation]
∴ `"dy"/"dx" = "v" + x * "dv"/"dx"`
∴ `"v" + x * "dv"/"dx" = (x^2 + "v"x^2 + "v"^2x^2)/x^2`
⇒ `"v" + x * "dv"/"dx" = (x^2(1 + "v" + v"^2))/x^2`
⇒ `"v" + x * "dv"/"dx" = 1 + "v" + "v"^2`
⇒ `x * "dv"/"dx" = 1 + "v" + "v"^2 - "v"`
⇒ `x * "dv"/"dx" = 1 + "v"^2`
⇒ `"dv"/(1 + "v"^2) = "dx"/x`
Integrating both sides, we get
`int "dv"/(1 + "v"^2) = int "dx"/x`
⇒ tan–1v = log x + c
⇒ `tan^-1 (y/x)` = log x + c
Hence, the required solution is `tan^-1 (y/x)` = log |x| + c.
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An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
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