solve the given differential equation by using an appropriate substitution. the de is of the form…

Explanation:

Step1: Make a substitution

Let $u = y - 2x+4$, then $\frac{du}{dx}=\frac{dy}{dx}-2$. The given differential - equation $\frac{dy}{dx}=2+\sqrt{y - 2x + 4}$ can be rewritten as $\frac{du}{dx}=\sqrt{u}$.

Step2: Separate variables

We have $\frac{du}{\sqrt{u}}=dx$.

Step3: Integrate both sides

Integrating $\int\frac{du}{\sqrt{u}}=\int dx$. Since $\int u^{-\frac{1}{2}}du = 2u^{\frac{1}{2}}+C_1$ and $\int dx=x + C_2$, we get $2\sqrt{u}=x + C$.

Step4: Substitute back

Substitute $u = y - 2x+4$ back into the equation, we have $2\sqrt{y - 2x + 4}=x + C$.

Answer:

$2\sqrt{y - 2x + 4}=x + C$