Answer to Question #169196 in Civil and Environmental Engineering for ezek

Question #169196

solve the follow homogenous equation of differential eq (2x-3y-1)dx (x 3y-5)dy=0


1
Expert's answer
2021-03-10T05:52:44-0500

We have given the differential equation,


"(2x-3y-1)dx + (x -3y-5)dy = 0"


"\\dfrac{dy}{dx} + \\dfrac{(2x -3y -1)}{(x -3y -5)} = 0\n"


Putting "x = X+h" and "y = Y+k"


"\\rightarrow 2h - 3k -1 = 0\\\\and\\text{ } h - 3k -5 = 0\n\n"


Hence, "\\dfrac{dY}{dX} + \\dfrac{(2X-3Y)}{(X+3Y)} = 0"


Putting "Y = vX"


"v+X\\dfrac{dv}{dX} + \\dfrac{2-3v}{1-3v} = 0\n"


"X\\dfrac{dv}{dX} = \\dfrac{(3v-2)}{(1-3v)} -v"


"= \\dfrac{(2v-2-3v^2)}{(1-3v)}"


"\\int(\\dfrac{1-3v}{2v-2-3v^2})dv + \\int\\dfrac{dX}{X} = logC"


"\\dfrac{1}{2}log(2v-2-3v^2)+logX=logc"


Hence, this is the final solution of the given differential equation.




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