AP Calculus BC 7.2 Verifying Solutions for Differential Equations - Exam Style Questions - MCQs - New Syllabus
Question
Which of the following is the solution to the differential equation \( \displaystyle \frac{dy}{dx}=x\,(y-2) \) with the initial condition \( y(2)=3 \)?
(A) \( y=2-e^{\,x^{2}/2}-2 \)
(B) \( y=2+e^{\,x^{2}/2-2} \)
(C) \( y=2+e^{\,2-x^{2}/2} \)
(D) \( y=3-e^{2}+e^{\,x^{2}/2} \)
(B) \( y=2+e^{\,x^{2}/2-2} \)
(C) \( y=2+e^{\,2-x^{2}/2} \)
(D) \( y=3-e^{2}+e^{\,x^{2}/2} \)
▶️ Answer/Explanation
Detailed solution
Separate: \( \dfrac{dy}{y-2}=x\,dx \).
Integrate: \( \ln|y-2|=\dfrac{x^{2}}{2}+C \Rightarrow y=2+C\,e^{x^{2}/2}. \)
Use \(y(2)=3\): \(1=C\,e^{2}\Rightarrow C=e^{-2}\).
So \( y=2+e^{\,x^{2}/2-2} \).
✅ Answer: (B)