Verifying Solutions for Differential Equations - AP Calculus AB Study Guide

The Core Idea: Verifying Solutions for Differential Equations

A differential equation is an equation that relates a function to its derivatives. The core idea of this topic is not to find the solution to a differential equation, but rather to verify if a given function is indeed a solution. This process is a direct check to see if the function "satisfies" the equation.

To verify, we substitute the given function and its calculated derivatives into the differential equation. If this substitution results in a true mathematical statement (i.e., the left side of the equation equals the right side), then the function is a solution. Some differential equations have a single, unique solution, while others have an entire family of solutions, often represented by a general solution containing a constant.

Key Definitions

This topic is centered on a process rather than specific formulas. The key definitions are:

• Differential Equation: An equation involving an unknown function and one or more of its derivatives. For example, $\frac{dy}{dx}=2y$ or $y^{\prime \prime }+4y=0$.

• Solution to a Differential Equation: A function $y=f(x)$ that, when substituted along with its derivatives into the differential equation, makes the equation a true statement.

• General Solution: A family of functions that serves as a solution to a differential equation. A general solution typically includes one or more arbitrary constants (e.g., $C$). For example, $y=C{e}^{2x}$ is the general solution to $\frac{dy}{dx}=2y$.

• Particular Solution: A single, unique function that is a solution to a differential equation. It is one specific case from the general solution, where the constant $C$ has been determined. For example, $y=5e^{2x}$ is a particular solution to $\frac{dy}{dx}=2y$.

Understanding the Verification Process

The central skill in this topic is the algebraic process of verification. It is fundamentally a substitution and simplification exercise. The goal is to prove that a given function works within the structure of a given differential equation.

The process is as follows:

1. Identify the components: Look at the differential equation and identify which derivatives of $y$ are needed (e.g., $y^{\prime }$, $y^{\prime \prime }$).

2. Differentiate: Take the necessary derivatives of the proposed solution function $y=f(x)$.

3. Substitute: Replace $y$ and its derivatives in the differential equation with the function and its calculated derivatives.

4. Simplify and Compare: Algebraically simplify both sides of the equation. If the left side is identical to the right side, the verification is complete, and the function is confirmed as a solution. If they are not identical, the function is not a solution.

This process is purely about checking, not solving. You are not expected to derive the solution from scratch, only to confirm that the one provided is correct.

Core Concepts & Rules

• A function $y=f(x)$ is a solution to a differential equation if it satisfies the equation when the function and its derivatives are substituted into it.

• The verification process involves finding the necessary derivatives of the proposed solution and plugging them into the differential equation.

• A differential equation may have a general solution, which represents an entire family of functions (e.g., $y=x^2+C$).

• A differential equation may have a particular solution, which is a single, unique function that satisfies the equation (e.g., $y=x^2+3$).

Step-by-Step Example 1: Verifying a Particular Solution

Problem: Verify that $y=4\sin (x)$ is a solution to the differential equation $\frac{d^{2}y}{d{x}^2}+y=0$.

Step 1: Find the necessary derivatives.

The differential equation involves $y$ and its second derivative, $y^{\prime \prime }$ (or $\frac{d^{2}y}{d{x}^2}$).

• Given function: $y=4\sin (x)$

• First derivative: $\frac{dy}{dx}=\frac{d}{dx}(4\sin (x))=4\cos (x)$

• Second derivative: $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(4\cos (x))=-4\sin (x)$

Step 2: Substitute the function and its second derivative into the differential equation.

The differential equation is $\frac{d^{2}y}{d{x}^2}+y=0$.

Substitute $\frac{d^{2}y}{d{x}^2}=-4\sin (x)$ and $y=4\sin (x)$:

$$(-4\sin (x))+(4\sin (x))=0$$

Step 3: Simplify and verify.

Simplify the left side of the equation:

$$-4\sin (x)+4\sin (x)=0$$

$$0=0$$

Since the left side equals the right side, the statement is true. Therefore, $y=4\sin (x)$ is a solution to the differential equation.

Step-by-Step Example 2: Verifying a General Solution

Problem: Show that any function of the form $y=\frac{1}{x+C}$, where $C$ is a constant, is a solution to the differential equation $\frac{dy}{dx}=-y^2$.

Step 1: Find the necessary derivative.

The differential equation involves $y$ and its first derivative, $\frac{dy}{dx}$.

• Given function: $y=(x+C{)}^{-1}$

• First derivative (using the Chain Rule): $\frac{dy}{dx}=-1(x+C{)}^{-2}\cdot (1)=-\frac{1}{(x+C{)}^2}$

Step 2: Substitute the function and its derivative into the differential equation.

The differential equation is $\frac{dy}{dx}=-y^2$.

• The Left Side (LS) is $\frac{dy}{dx}$, which we found to be $-\frac{1}{(x+C{)}^2}$.

• The Right Side (RS) is $-y^2$. Substitute $y=\frac{1}{x+C}$:

$$-y^2=-{\left(\frac{1}{x+C}\right)}^2=-\frac{1^2}{(x+C{)}^2}=-\frac{1}{(x+C{)}^2}$$

Step 3: Compare the left and right sides.

• LS: $-\frac{1}{(x+C{)}^2}$

• RS: $-\frac{1}{(x+C{)}^2}$

Since LS = RS, the equation holds true for any constant $C$. Therefore, $y=\frac{1}{x+C}$ is a general solution to the differential equation.

Using Your Calculator

Verifying solutions to differential equations is an analytical skill that must be performed by hand. The process relies on symbolic differentiation and algebraic simplification, which a calculator cannot perform for you in the way required on the AP Exam.

However, a graphing calculator can be a powerful tool for checking your work, specifically your derivatives.

How to check a derivative:

Suppose in Example 1, you calculated that the derivative of $y=4\sin (x)$ is y' = 4\cos(x)`. To check this: 1. In your calculator's graphing menu (`Y=`), enter your calculated derivative into `Y1`: `Y1 = 4cos(X)` 2. In `Y2`, use the numerical derivative function (`nDeriv` or $d/dx) to graph the derivative of the original function:

`Y2 = nDeriv(4sin(X), X, X)`

3. Graph both functions. If they produce the exact same graph (it will look like only one curve is drawn), your hand-calculated derivative is very likely correct. If you see two different graphs, you have made a differentiation error and should review your work before proceeding with the substitution step.

AP Exam Quick Hit

Common Question Types

• Multiple Choice Verification: You will be given a differential equation and several functions as options. You must test each option to see which one is a solution.

  • Example: "Which of the following is a solution to the differential equation $x{y}^{\prime }-y=x^2\cos (x)$?"

    • (A) $y=x\sin (x)$

    • (B) $y=x\cos (x)$

    • (C) $y=\sin (x)$

    • (D) $y=\cos (x)$

    (You would need to find $y^{\prime }$ for each option and substitute it and $y$ into the equation.)

• Free Response Part (a): The first part of a larger free-response question about differential equations often asks you to verify a given solution. This sets the stage for later parts of the question.

  • Example: "Consider the differential equation $\frac{dy}{dx}=(y-1{)}^2\cos (x)$. Show that $y=1-\frac{1}{\sin (x)+2}$ is a solution to the differential equation."

Common Mistakes

• Derivative Errors: The most frequent mistakes are simple errors made when calculating $y^{\prime }$ or $y^{\prime \prime }$. A mistake in applying the product, quotient, or chain rule will make it impossible to verify the solution correctly.

• Substitution Errors: Carelessly substituting the expressions for $y$ and its derivatives into the wrong places in the differential equation, or making sign errors during the substitution. For example, substituting $y$ for $y^{\prime }$.

• Algebraic Simplification Errors: After substituting correctly, students often make errors when simplifying the resulting expression, such as incorrect distribution, errors with fractions, or incorrect combination of terms.

• Confusing Verification with Solving: When asked to "verify" or "show that" a function is a solution, do not try to solve the differential equation from scratch. The task is only to plug in the provided function and prove that it works. Attempting to solve it wastes valuable time.