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

Explanation:

Step1: Rewrite the differential equation

The given differential equation is (-y\mathrm{d}x+(x + \sqrt{xy})\mathrm{d}y = 0), which can be rewritten as (\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{x+\sqrt{xy}}{y}). Since it is a homogeneous differential - equation, let (x = vy), then (\frac{\mathrm{d}x}{\mathrm{d}y}=v + y\frac{\mathrm{d}v}{\mathrm{d}y}).

Step2: Substitute (x = vy) into the differential equation

Substituting (x = vy) into (\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{x+\sqrt{xy}}{y}), we get (v + y\frac{\mathrm{d}v}{\mathrm{d}y}=\frac{vy+\sqrt{v y\cdot y}}{y}=v+\sqrt{v}).

Step3: Separate the variables

Subtracting (v) from both sides gives (y\frac{\mathrm{d}v}{\mathrm{d}y}=\sqrt{v}). Then, separating the variables, we have (\frac{\mathrm{d}v}{\sqrt{v}}=\frac{\mathrm{d}y}{y}).

Step4: Integrate both sides

Integrating (\int\frac{\mathrm{d}v}{\sqrt{v}}=\int\frac{\mathrm{d}y}{y}). We know that (\int\frac{\mathrm{d}v}{\sqrt{v}} = 2\sqrt{v}+C_1) and (\int\frac{\mathrm{d}y}{y}=\ln|y|+C_2). So, (2\sqrt{v}=\ln|y| + C) (where (C = C_2 - C_1)).

Step5: Substitute back (v=\frac{x}{y})

Substituting (v=\frac{x}{y}) into (2\sqrt{v}=\ln|y| + C), we get (2\sqrt{\frac{x}{y}}=\ln|y|+C).

Answer:

(2\sqrt{\frac{x}{y}}=\ln|y|+C)