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प्रश्न
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
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उत्तर १
`dy/dx = (x + 2y)/(x - y)`
Putting y = Vx
उत्तर २
The given differential equation is
`(x - y) dy/dx = x + 2y`
`=> dy/dx = (x + 2y)/(x - y)`
This is a homogeneous differential equation.
Putting y=vx and `dy/dx = v + x dy/dx`, we get
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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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