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Question
Show that the given differential equation is homogeneous and solve them.
`y' = (x + y)/x`
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Solution
`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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An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
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