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प्रश्न
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
उत्तर
(xy − x2) dy = y2 dx
Divide throughout by y2 (assuming y ≠ 0) to simplify:
`(xy-x^2)/y^2 dy = dx`
`(x/y - x^2/y^2) dy = dx`
To simplify, let us introduce a substitution. Divide the entire equation by x2 (assuming x ≠ 0):
`(1/x.y/x - 1/y) dy = 1/x^2 dx`
`v = y/x` (so that y = vx and dy = v dx + x dv)
Substitute into the equation:
`(1/xv - 1/v) (v dx + xdv) = 1/x^2 dx`
`(v/x - 1/(vx) (v dx + xdv) = 1/x^2 dx)`
Simplify step by step:
Multiply and separate:
`v^2/x dx + v/x xdv - 1/(vx) vdx - 1/(vx) xdv = 1/x^2 dx`
`v^2/x dx - 1/x dx + v dv - 1/vdv = 1/x^2 dx`
Group terms involving v and x. The equation becomes:
`(v^2 - 1)/v dv = (1/x^2 - (v^2-z)/x) dx`
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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)
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