Question

Solve dy/dx=x+2y-3/2x+y-3?

Answer 1

`y = 1/6x + 19/18 + Ce^{3x}`

Explanation:

Grouping like terms,

`dy/dx = 3y - 1/2x - 3`

In general, we want a differential equation of this type to follow the form `dy/dx + p(x)y = q(x)`.

`dy/dx - 3y = -1/2x - 3`

`color(red)(p(x) = -3)`

Multiply by the integrating factor `mu(x) = exp int color(red)(p(x))dx`.

`mu(x) = exp int color(red)(-3)dx = color(blue)(e^{-3x})`

`color(blue)(e^{-3x})dy/dx - 3color(blue)(e^{-3x})y = (-1/2x - 3)color(blue)(e^{-3x})`

Notice that the left hand side is the derivative of a product.

`d/dx(color(blue)(e^{-3x})y) = (-1/2x - 3)e^{-3x}`

Integrate both sides.

`int d/dx(color(blue)(e^{-3x})y) dx = -int(1/2x + 3)e^{-3x}dx`

`e^{-3x}y = 1/3(1/2x + 3)e^{-3x} + 1/18e^-3x + C`

`y = 1/3(1/2x + 3) + 1/18 + Ce^{3x}`

`y = 1/6x + 19/18 + Ce^{3x}`