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Question
Read the following passage:
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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)
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Solution
I. (x2 – y2) dx + 2xy dy = 0
`dy/dx = (y^2 - x^2)/(2xy)`
= `(x^2(y^2/x^2 - 1))/(x^2(2 y/x))`
= `g(y/x)`
II. Let y = vx
`dy/dx = v + x (dv)/dx`
`v + x (dv)/dx = (v^2x^2 - x^2)/(2vx^2)`
= `(v^2 - 1)/(2v)`
`x (dv)/dx = (v^2 - 1)/(2v) - v`
= `(v^2 - 1 - 2v^2)/(2v)`
= `-((1 + v^2))/(2v)`
`-int (2v)/(1 + v^2) dv = int dx/x`
`- log |1 + v^2| - log |x| + C` = 0
`- log |1 + y^2/x^2| - log |x| + C` = 0
`-log |(x^2 + y^2)/x^2| - log |x| + C` = 0
`- log |(x^2 + y^2)/x| + C` = 0
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