solve the given differential equation by using an appropriate substitution. the de is homogeneous. x dx+(y…

Explanation:

Step1: Rewrite the differential equation

Given (x dx+(y - 2x)dy = 0), we can rewrite it as (\frac{dx}{dy}=-\frac{y - 2x}{x}=-\frac{y}{x}+2). Let (v=\frac{x}{y}), then (x = vy) and (\frac{dx}{dy}=v + y\frac{dv}{dy}).

Step2: Substitute into the differential equation

Substituting (x = vy) and (\frac{dx}{dy}=v + y\frac{dv}{dy}) into (\frac{dx}{dy}=-\frac{y}{x}+2), we get (v + y\frac{dv}{dy}=-\frac{1}{v}+2).

Step3: Rearrange the equation

Rearrange to get (y\frac{dv}{dy}=-\frac{1}{v}+2 - v=\frac{-1 + 2v - v^{2}}{v}=\frac{-(v - 1)^{2}}{v}).

Step4: Separate variables

Separate variables: (\frac{v}{(v - 1)^{2}}dv=-\frac{dy}{y}).

Step5: Decompose the left - hand side fraction

Decompose (\frac{v}{(v - 1)^{2}}) using partial fractions. Let (\frac{v}{(v - 1)^{2}}=\frac{A}{v - 1}+\frac{B}{(v - 1)^{2}}). Then (v=A(v - 1)+B). Setting (v = 1) gives (B = 1), and comparing coefficients of (v) gives (A = 1). So (\frac{v}{(v - 1)^{2}}=\frac{1}{v - 1}+\frac{1}{(v - 1)^{2}}).

Step6: Integrate both sides

Integrate (\left(\frac{1}{v - 1}+\frac{1}{(v - 1)^{2}}\right)dv=-\frac{dy}{y}). (\int\frac{1}{v - 1}dv+\int\frac{1}{(v - 1)^{2}}dv=-\int\frac{dy}{y}). (\ln|v - 1|-\frac{1}{v - 1}=-\ln|y|+C).

Step7: Substitute back (v=\frac{x}{y})

(\ln\left|\frac{x}{y}-1\right|-\frac{1}{\frac{x}{y}-1}=-\ln|y|+C). (\ln\left|\frac{x - y}{y}\right|-\frac{y}{x - y}=-\ln|y|+C). (\ln|x - y|-\ln|y|-\frac{y}{x - y}=-\ln|y|+C). (\ln|x - y|-\frac{y}{x - y}=C).

Answer:

(\ln|x - y|-\frac{y}{x - y}=C)