Question

The solution of $\frac{dy}{dx} = (x + y)^2$ is

• A. $\tan^{-1}(x + y) = x + c$, where c is the constant of integration
• B. $x + y = \tan x + c$, where c is the constant of integration
• C. $x + y = \cot^{-1} x + c$, where c is the constant of integration
• D. $x + y = \sin^{-1}(x + y) + c$, where c is the constant of integration

Hint:

Always try the substitution $v = x+y$ when they appear together inside a function in a DE.

Answer 1

The Correct Option is A

Step 1: Concept
Use substitution for differential equations of form $\frac{dy}{dx} = f(ax+by+c)$.

Step 2: Meaning
Let $v = x + y$. Then $\frac{dv}{dx} = 1 + \frac{dy}{dx}$.

Step 3: Analysis
$\frac{dv}{dx} - 1 = v^2 \implies \frac{dv}{1+v^2} = dx$. Integrating both sides: $\tan^{-1}(v) = x + c$.

Step 4: Conclusion
Substituting $v$ back: $\tan^{-1}(x + y) = x + c$. Final Answer: (A)