The solution of the differential equation \( 9+y^{2}\displaystyle\frac{dy}{dx}=x^{2} \) is :
[where \( C \) is an arbitrary constant]
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( x^{3}-y^{3}=27x+C \)
The variables separate:
\( \displaystyle y^{2}\,dy=(x^{2}-9)\,dx \)
Integrating both sides,
\( \displaystyle \frac{y^{3}}{3}=\frac{x^{3}}{3}-9x+C_{1} \)
Multiplying by 3,
\( \displaystyle y^{3}=x^{3}-27x+3C_{1} \)
\( \displaystyle x^{3}-y^{3}=27x+C \)
Here \( C=-3C_{1} \) is again an arbitrary constant.