Question

solve the given differential equation by using an appropriate substitution. the de is of the form $\frac{dy}{dx}=f(ax + by + c)$. $\frac{dy}{dx}=(x + y+3)^2$

Explanation:

Step1: Make a substitution

Let \(u=x + y+3\), then \(\frac{du}{dx}=1+\frac{dy}{dx}\), and the given differential equation \(\frac{dy}{dx}=(x + y + 3)^2\) can be rewritten as \(\frac{du}{dx}-1 = u^2\).

Step2: Rearrange the equation

We get \(\frac{du}{dx}=u^2 + 1\). 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

Substitute \(u=x + y+3\) back into the equation, we have \(\arctan(x + y+3)=x + C\). Then \(x + y+3=\tan(x + C)\), and \(y=\tan(x + C)-x - 3\).

Answer:

\(y=\tan(x + C)-x - 3\)