solve the given differential equation $\frac{dy}{dx}=(x + y+5)^{2}$

Explanation:

Step1: Use substitution

Let (u=x + y+5), then (\frac{du}{dx}=1+\frac{dy}{dx}), and the given differential equation (\frac{dy}{dx}=(x + y + 5)^2) becomes (\frac{du}{dx}-1 = u^2).

Step2: Rearrange the equation

We can rewrite (\frac{du}{dx}-1 = u^2) as (\frac{du}{dx}=u^2 + 1). Then separate the variables: (\frac{du}{u^2+1}=dx).

Step3: Integrate both sides

Integrating (\int\frac{du}{u^2 + 1}=\int dx). We know that (\int\frac{du}{u^2+1}=\arctan(u)+C_1) and (\int dx=x + C_2). So (\arctan(u)=x + C), where (C = C_2 - C_1).

Step4: Substitute back (u)

Substitute (u=x + y+5) back into the equation, we get (\arctan(x + y+5)=x + C).

Step5: Solve for (y)

(x + y+5=\tan(x + C)), then (y=\tan(x + C)-x - 5).

Answer:

(y=\tan(x + C)-x - 5)