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Question
Solve the differential equation: ` (dy)/(dx) = (x + y )/ (x - y )`
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Solution
The given differential equation is:
⇒ `dy/dx = (x + y)/( x - y)` ....(1)
Let F (x, y) = `(x + y)/( x - y)`
∴ F ( λx, λy) = `(λx + λy)/( λx - λy) = (x + y)/( x - y) = λ° . F(x, y)`
Thus, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as: y = vx
⇒ `d/dx (y) = d/dx (vx)`
⇒ `dy/dx = v + x (dv)/dx`
Substituting the values of y and in equation (1), we get:
`v + x (dv)/(dx) = (x + vx)/(x - vx) = (1 + v)/(1 - v)`
⇒ `x (dv)/(dx) = (1 + v)/(1 - v) - v = (1 + v - v( 1 - v))/( 1 - v)`
⇒ `x (dv)/(dx) = (1 + v^2)/(1 - v)`
⇒ `(1 - v)/(1 + v^2) (dv) = (dx)/x`
Integrating both sides, we get:
`tan^-1v - 1/2 log ( 1 + y^2 ) = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log [ 1 + (y/x)^2 ] = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log ((x^2 + y^2)/x^2) = log x + c`
⇒ `tan^-1 (y/x) - 1/2 [ log ((x^2 + y^2)- log x^2) ] = log x + c`
⇒ `tan^-1 (y/x) - 1/2 log (x^2 + y^2) + c`
This is the required solution of the given differential equation.
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