Show that the given differential equation is homogeneous and solve them. y′=x+yx - Mathematics

प्रश्न

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

`y' = (x + y)/x`

बेरीज

उत्तर

`dy/dx = (x + y)/x`

∵ The degree of numerator and denominator is the same so the given differential equation is a homogeneous differential equation.

∴ Putting y = vx

In equation (i)

`dy/dx = v + x dy/dx`

`v + x .dy/dx = (x + vx)/x`

`v + x .dy/dx = 1 + v`

`=> x. dy/dx = 1`

`=> dv = dx/x`

On integrating on both sides,

`int 1. (dv) = int 1/x` dv

v = log `abs x = C`

`=> y/x = log abs x + C`

`= y = x log abs x + Cx`

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

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Q 2 Q 1 Q 3

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 2 | पृष्ठ ४०६

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