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Question
Solve the following differential equation:
`x^2. dy/dx = x^2 + xy + y^2`
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Solution
`x^2. dy/dx = x^2 + xy + y^2`
∴ `dy/dx = (x^2 + xy + y^2)/x^2` ...(1)
Put y = vx
∴ `dy/dx = v + x (dv)/dx`
∴ (1) becomes, `v + x (dv)/dx = (x^2 + x * vx + v^2x^2)/x^2`
∴ `v + x (dv)/dx = 1 + v + v^2`
∴ `x (dv)/dx = 1 + v^2`
∴ `1/(1 + v^2) dv = 1/x dx`
Integrating, we get
`int 1/(1 + v^2) dv = int 1/x dx + c`
∴ tan–1 v = log |x| + c
∴ `tan^-1 (y/x) = log |x| + c`
This is the general solution.
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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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