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Question
Solve the following differential equation:
`x * dy/dx - y + x * sin(y/x) = 0`
Solution 1
`"x" "dy"/"dx" - "y" + "x sin"("y"/"x") = 0` ...(1)
Put y = vx
∴ `"dy"/"dx" = "v + x" "dv"/"dx" and "y"/"x" = "v"`
∴ equation (1) becomes,
`x("v + x""dv"/"dx") - "vx + x sin v" = 0`
∴ `"vx" + "x"^2 "dv"/"dx" - "vx" + "x sin v" = 0`
∴ `"x"^2 "dv"/"dx" + "x sin v" = 0`
∴ `1/"sin v" "dv" + 1/"x" "dx" = 0`
Integrating, we get
∴ `int "cosec v dv" + int1/"x" "dx" = "c"_1`
∴ `log |"cosec v - cot v"| + log |"x"| = log "c"`, where c1 = log c
∴ `log |"x" ("cosec v" - "cot v")| = log "c"`
∴ `"x"(1/(sin"v") - (cos "v")/(sin"v")) = "c"`
∴ x(1 - cos v) = c sin v
∴ `"x"[1 - cos("y"/"x")] = "c sin"("y"/"x")`
This is the general solution.
Solution 2
`x * dy/dx - y + x * sin(y/x) = 0`
Put y = vx
⇒ `dy/dx = v + x (dv)/dx`
∴ The given equation becomes,
`x (v + x (dv)/dx) - vx + x sin v = 0`
∴ `vx + x^2 (dv)/dx - vx + x sin v = 0`
∴ `x^2 (dv)/dx = -x sin v`
∴ `(dv)/(sin v) = - dx/x` ...(variable separable form)
Integrating on both sides, we get,
`int cosec v dv = -int 1/x dx`
∴ log (cosec v − cot v) = − log x + log c
∴ log (cosec v − cot v) = `log (c/x)`
∴ cosec v − cot v = `c/x`
∴ `x [cosec (y/x) - cot (y/x)] = c` is the required solution.
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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. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
