7.6 Finding Particular Solutions Using Initial Conditions

• A general solution to a differential equation satisfies the equation and includes a constant of integration.


  True
• Differential equations are equations that involve derivatives
• Initial conditions are used to determine the constant of integration in a particular solution.


  True
• 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
• 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
• Different methods to solve differential equations are suited for specific types of equations.


  True
• A general solution to a differential equation includes an arbitrary constant of integration denoted by $C$.


  True
• The method of separation of variables is used when the equation can be separated into f(y)dy = g(x)dx
• What is obtained by integrating a differential equation?

  General solution
• 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.
• A differential equation involves derivatives.
• 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
• What do initial conditions specify?

  Value of the function
• 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
• What does the resulting expression after integration represent?

  General solution
• Which method involves rearranging the equation to separate variables and integrate both sides?

  Separation of Variables
• A particular solution has a specific value for the constant $C$.


  True
• A general solution includes an arbitrary constant denoted by C.
• 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
• The particular solution $y =$$x^{2} +$$3$ satisfies the differential equation $\frac{dy}{dx} =$$2x$ with the initial condition $y(0) =$$3$


  True
• The general solution of a differential equation always includes an arbitrary constant $C$


  True
• Match the term with its definition:

  General Solution ↔️ Family of solutions to the differential equation

  Particular Solution ↔️ Specific solution satisfying the initial condition
• When verifying a particular solution, you must substitute it into the left-hand side of the differential equation


  True
• 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
• What is the first step to verify the particular solution $y =$$x^{2} +$$3$ for the differential equation $\frac{dy}{dx} =$$2x$?


  Substitute the solution
• The particular solution $y =$$x^{2} +$$3$ satisfies the differential equation \frac{dy}{dx} = 2x</latex>


  True
• An initial condition specifies the value of the function at a given point.


  True
• What is the particular solution of the differential equation $\frac{dy}{dx} =$$2x$ with y(0) = 3</latex>?


  $y =$$x^{2} +$$3$
• A particular solution is a specific function that satisfies both the differential equation and the initial condition.


  True
• In the verification process, the left-hand side of the equation must match the right-hand side.
• Match the term with its definition:

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

  Particular Solution ↔️ A specific solution obtained by using an initial condition

  Initial Condition ↔️ A condition that specifies the value of the function at a given point
• A particular solution is determined by initial conditions.
• The constant $C$ in a particular solution is always arbitrary.


  False
• What is the value of $C$ in the particular solution $y =$$x^{2} +$$3$?


  $3$
• A particular solution to a differential equation is found by using an initial condition
• 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