Show that the given differential equation is homogeneous and solve them. x dy-y dx= x2+y2 dx - Mathematics

प्रश्न

Show that the given differential equation is homogeneous and solve them.

`x dy - y dx = sqrt(x^2 + y^2) dx`

बेरीज

उत्तर

`x dy - y dx = sqrt(x^2 + y^2) dx`,

Which can be written as `x dy/dx = y + sqrt (x^2 + y^2)`

or `dy/dx = y/x + sqrt (1 + (y/x)^2)` ....(1)

Since R.H.S. is of the form `g(y/x)`, and so it is a homogeneous function of degree zero.

Therefore equation (1) is a homogeneous differential equation.

To solve this, put y = vx

⇒ `dy/dx = v + x (dv)/dx`

Substituting the value of y and `dy/dx` in (1), we get

`v + x (dv)/dx = v + sqrt (1 + v^2)`

⇒ `x (dv)/dx = sqrt(1 + v^2)`

⇒ `dx/x = (dv)/sqrt(1 + v^2)`

⇒ `int dx/x = int (dv)/ sqrt(1 + v^2)`

⇒ `log x + log C_1 = log |v + sqrt (1+ v^2)|`

⇒ `log x + log C_1 = log |y/x + sqrt (1 + y^2/x^2)|`

⇒ `log C_1 x = log |y + sqrt (x^2 + y^2)| - log x`

⇒ `pm C_1 x^2 = y + sqrt (x^2 + y^2)`

⇒ `Cx^2 = y + sqrt (x^2 + y^2)`

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Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

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Q 6 Q 5 Q 7

पाठ 9: Differential Equations - Exercise 9.5 [पृष्ठ ४०६]

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एनसीईआरटी Mathematics Part 1 and 2 [English] Class 12

पाठ 9 Differential Equations
Exercise 9.5 | Q 6 | पृष्ठ ४०६

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