Question

If $x(1+y^{2})\,dx + y(1+x^{2})\,dy = 0$ and $y(0) = 1$, then $x^{2}y^{2} + x^{2} + y^{2}$ equals

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

After integrating and finding the general solution, always substitute the initial condition to determine the constant before simplifying the expression.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Separate variables and integrate, then apply the initial condition.
Step 2: Detailed Explanation:
Divide by $(1+x^2)(1+y^2)$: $\dfrac{x}{1+x^2}dx + \dfrac{y}{1+y^2}dy = 0$.
Integrate: $\dfrac{1}{2}\ln(1+x^2) + \dfrac{1}{2}\ln(1+y^2) = C$.
At $y(0)=1$: $0 + \dfrac{1}{2}\ln 2 = C$.
$(1+x^2)(1+y^2) = 2 \Rightarrow 1 + x^2 + y^2 + x^2y^2 = 2 \Rightarrow x^2y^2 + x^2 + y^2 = 1$.
Step 3: Final Answer:
$x^2y^2 + x^2 + y^2 = 1$.