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

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Answer 1 · 2018-08-11T15:41:02

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

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}#