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Solve the Differential Equation: D Y D X = X + Y X − Y - Mathematics

Sum

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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