7.6 Finding Particular Solutions Using Initial Conditions

• What is the particular solution for \(y' = 2x\) with the initial condition \(y(0) = 3\)?

  $y =$$x^{2} +$$3$
• What does a general solution include?

  A constant of integration
• The general solution of a differential equation provides a family of solutions
• Match the method with its example for finding general solutions:

  Separation of Variables ↔️ $y =$$x^{2} +$$C$

  Integrating Factors ↔️ $y =$$\frac{1}{2}e^{x} +$$Ce^{ - x}$
• A particular solution provides a unique function because the value of C is determined using an initial condition.


  True
• In the separation of variables method, variables are separated on each side of the equation before integration.


  True
• The purpose of substituting initial conditions is to find the particular
• Steps to solve practical problems using initial conditions

  1️⃣ Formulate the differential equation

  2️⃣ Find the general solution

  3️⃣ Apply the initial condition

  4️⃣ Write the particular solution

  5️⃣ Interpret the solution in context
• A particular solution satisfies an initial condition.


  True
• Solving a differential equation means finding the function that satisfies the equation
• What does an initial condition determine in the general solution of a differential equation?

  Constant of integration
• A general solution to a differential equation is a family of functions that includes a constant of integration
• What two conditions must a particular solution satisfy?

  Differential equation and initial condition
• The general solution includes a constant of integration
• Substituting initial conditions determines the value of C in the general solution.


  True
• What role do initial conditions play in solving practical problems modeled by differential equations?

  They determine the unique solution
• A general solution of a differential equation includes a constant of integration called C
• Why is a general solution not unique?

  It includes a constant of integration
• What is the purpose of an initial condition?

  To determine the constant of integration
• What is the purpose of an initial condition in finding a particular solution?

  Determine the constant of integration
• A differential equation relates a function with its derivatives.


  True
• What is the general solution to a differential equation?

  A family of functions with C
• Steps to find the particular solution using initial conditions

  1️⃣ Obtain the general solution

  2️⃣ Plug in the initial condition

  3️⃣ Solve for C

  4️⃣ Replace C in the general solution
• The particular solution y = x^2 + 3 satisfies the differential equation y' = 2x.


  True
• What is a differential equation?

  Relates a function with its derivatives
• A particular solution is a family of functions.

  False
• Match the type of solution with its description:

  General Solution ↔️ A family of functions with constant C

  Particular Solution ↔️ A specific function satisfying initial condition
• A particular solution satisfies both the differential equation and the initial condition.


  True
• What is the key difference between a general solution and a particular solution?

  Particular solution satisfies initial condition
• Steps for solving a differential equation using Separation of Variables:

  1️⃣ Separate variables on each side

  2️⃣ Integrate both sides

  3️⃣ Solve for y
• What is an integrating factor used for in solving differential equations?

  To make the equation exact
• How is a particular solution verified?

  By substituting into the equation
• In the example of exponential growth, the initial population is used to find the constant A in the general solution.


  True