AP Calculus BC 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables - Exam Style Questions - MCQs - New Syllabus
No-CalcQuestion
Let \( y=f(x) \) satisfy \( \dfrac{dy}{dx}=3x^{2}y \) with \( f(0)=6 \). Which of the following is \( f(x) \)?
(A) \( e^{x^{3}} \)
(B) \( e^{x^{3}}+5 \)
(C) \( e^{x^{3}}+6 \)
(D) \( 6e^{x^{3}} \)
(B) \( e^{x^{3}}+5 \)
(C) \( e^{x^{3}}+6 \)
(D) \( 6e^{x^{3}} \)
▶️ Answer/Explanation
Detailed solution
\[ \frac{1}{y}\,dy=3x^{2}\,dx \ \Rightarrow\ \ln|y|=x^{3}+C \ \Rightarrow\ y=Ae^{x^{3}}. \] From \( f(0)=6 \) we get \( A=6 \), so \( y=6e^{x^{3}} \). ✅ Correct: (D)
No-CalcQuestion
Which of the following is the solution to the differential equation \(\dfrac{dy}{dx}=-3y\) with the initial condition \(y(0)=7\)?
(A) \(y=7e^{-3x}\)
(B) \(y=e^{-3x}+6\)
(C) \(y=-\dfrac{3}{2}x^2+7\)
(D) \(y=\sqrt{49-6x}\)
(B) \(y=e^{-3x}+6\)
(C) \(y=-\dfrac{3}{2}x^2+7\)
(D) \(y=\sqrt{49-6x}\)
▶️ Answer/Explanation
Correct answer: (A) \(y=7e^{-3x}\)
Separate variables: \(\dfrac{1}{y}\,dy=-3\,dx=-3\times dx\). Integrate: \(\ln|y|=-3\times x+C\Rightarrow y=C e^{-3x}\).
Use \(y(0)=7\Rightarrow C=7\). So \(y=7e^{-3x}\).