Question

solve the given differential equation by using an appropriate substitution. the de is homogeneous.
(x - y) dx + x dy = 0

Explanation:

Step1: Rearrange the differential equation

We rewrite $(x - y)dx+xdy = 0$ as $xdy=-(x - y)dx$, then $\frac{dy}{dx}=\frac{y - x}{x}=\frac{y}{x}-1$.

Step2: Use substitution

Let $v=\frac{y}{x}$, so $y = vx$. Differentiating $y$ with respect to $x$ using the product - rule, we get $\frac{dy}{dx}=v + x\frac{dv}{dx}$.
Substitute $\frac{dy}{dx}$ and $y$ into the differential equation: $v + x\frac{dv}{dx}=v - 1$.

Step3: Solve the new differential equation

Simplify the equation $v + x\frac{dv}{dx}=v - 1$ to get $x\frac{dv}{dx}=-1$.
Separate the variables: $dv=-\frac{1}{x}dx$.
Integrate both sides: $\int dv=-\int\frac{1}{x}dx$.
We know that $\int dv = v+C_1$ and $\int\frac{1}{x}dx=\ln|x|+C_2$. So $v=-\ln|x|+C$.

Step4: Back - substitute

Since $v = \frac{y}{x}$, we have $\frac{y}{x}=-\ln|x|+C$.
Then $y=-x\ln|x|+Cx$.

Answer:

$y=-x\ln|x|+Cx$