7.2 Verifying Solutions for Differential Equations

• What is a differential equation?

  Relates a dependent variable to its derivatives
• The solution type for a differential equation is a function
• The general form of a differential equation is \frac{d^{n} y}{dx^{n}}</latex>
• An algebraic equation involves derivatives, while a differential equation does not

  False
• In a differential equation, $x$ is referred to as the independent variable.
• What is the term for the variable $x$ in a differential equation?


  Independent variable
• Algebraic equations involve derivatives, while differential equations do not.

  False
• Algebraic equations relate constants and variables without derivatives.


  True
• When identifying a potential solution, one should consider basic functions such as polynomials, exponentials, and trigonometric functions.
• What is the first step in substituting a potential solution into a differential equation?

  Calculate derivatives
• If substituting a potential solution into a differential equation results in both sides being equal, the solution is verified.

  True
• Unlike algebraic equations, differential equations find functions that satisfy specific derivative relationships.
• A differential equation involves derivatives of the dependent variable $y$ with respect to the independent variable x
• Give an example of a differential equation.

  $\frac{dy}{dx} =$$2x +$$y$
• Steps to identify a potential solution for a differential equation:

  1️⃣ Examine the equation's terms

  2️⃣ Consider basic functions

  3️⃣ Make an educated guess
• An actual solution is thoroughly validated and correct.


  True
• What is the derivative of y = x^{2} + C</latex> with respect to $x$?


  $\frac{dy}{dx} =$$2x$
• What does combining like terms involve in simplification?

  Grouping similar terms
• After substituting a potential solution, both sides of the equation must be checked for equality.
• What is the first step in simplifying an equation for a differential equation solution?

  Combining like terms
• Factoring involves decomposing expressions to reveal common elements.


  True
• Steps for simplifying after substituting a potential solution

  1️⃣ Combine like terms

  2️⃣ Apply algebraic identities

  3️⃣ Factor expressions
• What is the fundamental difference between a differential equation and an algebraic equation?

  Derivative relationships
• Match the feature with the correct type of equation:

  Variables are dependent and independent ↔️ Differential Equation

  Solution type is numerical values ↔️ Algebraic Equation
• $\frac{d^{n} y}{dx^{n}}$ is the nth-order derivative of the dependent variable $y$ with respect to the independent variable x
• What is the first step in identifying a potential solution for a differential equation?

  Examine the terms
• Steps to substitute a potential solution into a differential equation:

  1️⃣ Calculate the necessary derivatives

  2️⃣ Simplify the equation

  3️⃣ Check for equality
• Steps to substitute a potential solution into a differential equation:

  1️⃣ Calculate the necessary derivatives

  2️⃣ Replace the terms in the equation

  3️⃣ Simplify the resulting equation

  4️⃣ Check for equality
• After substitution, the next step is to simplify the equation by combining like terms, applying algebraic identities, and factoring expressions.
• If the simplified expressions after substitution are equal, the potential solution is a true solution.


  True
• Match the simplification technique with its purpose:

  Combining Like Terms ↔️ Reduce complexity and facilitate comparison

  Applying Algebraic Identities ↔️ Rewrite expressions in simpler forms

  Factoring ↔️ Reveal common elements and simplify equations
• To determine if a potential solution is a true solution, it is necessary to check for equality after simplifying the equation.
• A differential equation involves both dependent and independent variables


  True
• What is the solution to the differential equation $\frac{dy}{dx} =$$2x$?


  $y =$$x^{2} +$$C$
• What does $\frac{d^{n} y}{dx^{n}}$ represent in the general form of a differential equation?


  nth-order derivative
• What type of relationships do differential equations describe?

  Derivative relationships
• What is the purpose of substituting a potential solution into a differential equation?

  Verify if it is a true solution
• The function f(x, y, \frac{dy}{dx}, \frac{d^{2}y}{dx^{2}}, ..., \frac{d^{n - 1}y}{dx^{n - 1}})</latex> depends on the independent variable $x$, the dependent variable $y$, and the derivatives of $y$ up to order n-1
• The general form of a differential equation is $\frac{d^{n} y}{dx^{n}} =$$f(x, y, \frac{dy}{dx}, \frac{d^{2}y}{dx^{2}}, ..., \frac{d^{n - 1}y}{dx^{n - 1}})$, where $\frac{d^{n} y}{dx^{n}}$ represents the nth-order derivative
• What is the first step in identifying a potential solution for a differential equation?

  Examine the equation's terms