State the type of the differential equation for the equation. xdy – ydx = dx2+y2 dx and solve it

Question

State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it

Sum

Solution

Given equation can be written as xdy = `(sqrt(x^2 + y^2) + y) "d"x`

i.e., `"dy"/"dx" = (sqrt(x^2 + y^2) + y)/x` ......(1)

Clearly R.H.S of (1) is a homogeneous function of degree zero.

Therefore, the given equation is a homogeneous differential equation.

Substituting y = vx, we get from (1)

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

i.e. `"v" + x "dv"/"dx" = sqrt(1 + "v"^2) + "v"`

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

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

Integrating both sides of (2), we get

`log("v" + sqrt(1 + "v"^2))` = logx + logc

⇒ `"v" + sqrt(1 + "v"^2)` = cx

⇒ `y/x + sqrt(1 + y^2/x^2)` = cx

⇒ `y + sqrt(x^2 + y^2)` = cx²

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Methods of Solving Differential Equations> Homogeneous Differential Equations

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Q 11 Q 10 Q 12

Chapter 9: Differential Equations - Solved Examples [Page 186]

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NCERT Exemplar Mathematics Exemplar [English] Class 12

Chapter 9 Differential Equations
Solved Examples | Q 11 | Page 186

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