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

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

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

Answer

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)