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AP Calculus BC 7.2 Verifying Solutions for Differential Equations Study Notes - New Syllabus

AP Calculus BC 7.2 Verifying Solutions for Differential Equations Study Notes- New syllabus

AP Calculus BC 7.2 Verifying Solutions for Differential Equations Study Notes – AP Calculus BC-  per latest AP Calculus BC Syllabus.

LEARNING OBJECTIVE

  • Solving differential equations allows us to determine functions and develop models.

Key Concepts: 

  • Verifying Solutions for Differential Equations

AP Calculus BC-Concise Summary Notes- All Topics

AP Calculus BC Exam Questions MCQs and FRQs

Verifying Solutions for Differential Equations

Verifying Solutions for Differential Equations

Verifying that a given function is a solution to a differential equation involves checking whether the function and its derivatives satisfy the equation for all values in its domain.

Key Points:

  • A differential equation is an equation involving an unknown function \( y = f(x) \) and its derivatives such as \( \dfrac{dy}{dx} \), \( \dfrac{d^2y}{dx^2} \), etc.
  • A solution is a function that, when substituted into the differential equation, satisfies it for all \( x \) in the domain.
  • There may be infinitely many solutions due to the presence of constants of integration.
  • Solutions are classified as:
    • General solution: Contains arbitrary constants, e.g. \( y = Ce^{2x} \).
    • Particular solution: Obtained by assigning specific values to constants based on initial/boundary conditions.

Mathematical Form:

  • First-order DE: \( F\left(x, y, \dfrac{dy}{dx}\right) = 0 \)
  • Second-order DE: \( F\left(x, y, \dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right) = 0 \)

Steps to Verify a Solution:

  1. Compute the required derivative(s) of the given function \( y = f(x) \).
  2. Substitute \( y \), \( \dfrac{dy}{dx} \), \( \dfrac{d^2y}{dx^2} \), etc., into the given DE.
  3. Simplify the resulting expression.
  4. If the left-hand side (LHS) equals the right-hand side (RHS) for all \( x \) in the domain, the function is a valid solution.

Example Idea:

If the DE is \( \dfrac{dy}{dx} = 3y \) and the given function is \( y = Ce^{3x} \):

  • Derivative: \( \dfrac{dy}{dx} = 3Ce^{3x} \)
  • Substitute: LHS \( = 3Ce^{3x} \), RHS \( = 3y = 3Ce^{3x} \)
  • Since LHS = RHS for all \( x \), it is a solution.

Example:

Verify that \( y = e^{2x} \) is a solution of the differential equation:

\(\dfrac{dy}{dx} = 2y\)

▶️Answer/Explanation

Find the derivative of \( y \). \( y = e^{2x} \) \( \dfrac{dy}{dx} = 2e^{2x} \)

Substitute \( y \) and \( \dfrac{dy}{dx} \) into the DE: LHS: \( \dfrac{dy}{dx} = 2e^{2x} \) RHS: \( 2y = 2(e^{2x}) = 2e^{2x} \)

Since LHS = RHS, the function satisfies the DE. 

Example:

Verify that \( y = C_1\cos(3x) + C_2\sin(3x) \) is a solution of the differential equation:

\(\dfrac{d^2y}{dx^2} + 9y = 0\)

▶️Answer/Explanation

 First derivative: \( \dfrac{dy}{dx} = -3C_1\sin(3x) + 3C_2\cos(3x) \)

 Second derivative: \( \dfrac{d^2y}{dx^2} = -9C_1\cos(3x) – 9C_2\sin(3x) \)

 Substitute into the DE: LHS: \( \dfrac{d^2y}{dx^2} + 9y = (-9C_1\cos(3x) – 9C_2\sin(3x)) + 9(C_1\cos(3x) + C_2\sin(3x)) = 0 \) Since the equation holds for all \( x \), the function is a valid solution. 

Example:

Verify that \( y = Ae^{kt} \) is a solution to the differential equation:

\(\dfrac{dy}{dt} = ky\)

▶️Answer/Explanation

 Derivative: \( \dfrac{dy}{dt} = Ake^{kt} \)

 RHS: \( ky = k(Ae^{kt}) = Ake^{kt} \) LHS = RHS, so the function satisfies the DE. 

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