7.6 Finding Particular Solutions Using Initial Conditions

• Differential equations are equations that involve derivatives
• A general solution to a differential equation satisfies the equation and includes a constant of integration.


  True
• A particular solution to a differential equation is found by using an initial condition
• Match the term with its definition:

  Differential Equation ↔️ An equation involving derivatives

  General Solution ↔️ Family of functions satisfying the equation

  Particular Solution ↔️ Specific solution obtained by initial condition

  Initial Condition ↔️ Specifies function value at a point
• Initial conditions are used to determine the constant of integration in a particular solution.


  True
• To find the particular solution of the differential equation $\frac{dy}{dx} =$$2x$ with $y(0) =$$3$, the value of the constant $C$ is 3
• The method of separation of variables is used when the equation can be separated into f(y)dy = g(x)dx
• Steps to find a particular solution using an initial condition:

  1️⃣ Find the general solution

  2️⃣ Apply the initial condition

  3️⃣ Solve for the constant $C$
• To find a particular solution, the values from the initial condition are substituted into the general solution.
• The particular solution of the differential equation $\frac{dy}{dx} =$$2x$ with the initial condition $y(0) =$$3$ is y = x^{2} + 3</latex>.


  True
• What does a general solution represent?

  A family of functions
• Match the term with its definition:

  Differential Equation ↔️ An equation involving derivatives of a function

  General Solution ↔️ The family of functions that satisfy the differential equation

  Particular Solution ↔️ A specific solution obtained by using an initial condition
• Initial conditions are used to find a particular solution.
• Which method involves rearranging the equation to separate variables and integrate both sides?

  Separation of Variables
• A general solution includes an arbitrary constant denoted by C.
• Match the type of solution with its definition:

  General Solution ↔️ Family of functions satisfying the differential equation

  Particular Solution ↔️ Specific solution determined by initial conditions
• What is the first step in finding a particular solution to a differential equation using initial conditions?

  Obtain general solution
• Match the term with its definition:

  General Solution ↔️ Family of solutions to the differential equation

  Particular Solution ↔️ Specific solution satisfying the initial condition
• Steps to find a particular solution using an initial condition:

  1️⃣ Obtain the general solution

  2️⃣ Substitute the initial condition

  3️⃣ Solve for the constant $C$

  4️⃣ Replace $C$ in the general solution
• What is the particular solution for the differential equation $\frac{dy}{dx} =$$2x$ with the initial condition $y(0) =$$3$?


  $y =$$x^{2} +$$3$
• Match the term with its definition:

  General Solution ↔️ Family of solutions to the differential equation

  Particular Solution ↔️ Specific solution satisfying the initial condition
• Steps to verify a particular solution:

  1️⃣ Substitute the particular solution into the original equation

  2️⃣ Evaluate the necessary derivatives

  3️⃣ Compare both sides of the equation
• When verifying a particular solution, calculate the necessary derivatives of the solution.
• The derivative of $y =$$x^{2} +$$3$ is 2x</latex>


  True
• In the verification process, the left-hand side of the equation must match the right-hand side.
• What is a differential equation?

  Equation with derivatives
• Different methods to solve differential equations are suited for specific types of equations.


  True
• Match the method to solve differential equations with its description:

  Separation of Variables ↔️ Rearrange and integrate variables

  Integrating Factors ↔️ Multiply by a factor to make exact

  Laplace Transforms ↔️ Convert to algebraic equation

  Numerical Methods ↔️ Approximate solutions
• A general solution to a differential equation includes an arbitrary constant of integration denoted by $C$.


  True
• What is obtained by integrating a differential equation?

  General solution
• What is the next step after substituting the initial condition into the general solution?

  Solve for the constant
• A differential equation involves derivatives.
• A particular solution is found by using an initial condition to determine the constant of integration.

  True
• What do initial conditions specify?

  Value of the function
• Steps to use initial conditions to find a particular solution

  1️⃣ Find the general solution of the differential equation.

  2️⃣ Substitute the initial condition into the general solution.

  3️⃣ Solve for the constant of integration.

  4️⃣ Replace the constant in the general solution to obtain the particular solution.
• The method of separation of variables requires the equation to be separable into the form $f(y)dy =$$g(x)dx$.


  True
• What does the resulting expression after integration represent?

  General solution
• A particular solution has a specific value for the constant $C$.


  True
• The particular solution for the differential equation $\frac{dy}{dx} =$$2x$ with $y(0) =$$3$ is $y =$$x^{2} +$$3$


  True
• What is the purpose of using an initial condition in differential equations?

  Find a particular solution