Question

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:

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