7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

• A particular solution is obtained from the general solution by using given initial conditions
• The particular solution to $\frac{dy}{dx} =$$2x$ with initial condition $y(0) =$$1$ is $y =$$x^{2} +$$1$.
• What is a particular solution in the context of differential equations?

  A solution with initial conditions
• Initial conditions are used to determine the arbitrary constants in the general solution.
• For $\frac{dy}{dx} =$$2x$ with general solution $y =$$x^{2} +$$C$, the initial condition $y(0) =$$1$ gives $C =$$1$, so the particular solution is y = x^{2} + 1
• A particular solution is a single function with no undefined constants.
• Separation of Variables is a technique used to solve differential equations where $\frac{dy}{dx}$ can be expressed as a product of functions of $x$ and $y$, allowing $y$ terms to be on one side and $x$ terms on the other
• Steps for solving differential equations using Separation of Variables

  1️⃣ Separate Variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$

  4️⃣ Apply Initial Conditions
• Solve $\frac{dy}{dx} =$$xy$ with $y(0) =$$2$ using Separation of Variables.


  $y =$$2e^{\frac{1}{2}x^{2}}$
• What is the method of separation of variables used for?

  Solving differential equations
• The method of separation of variables expresses $\frac{dy}{dx}$ as a product of functions of x and $y$
• The first step in separation of variables is to rearrange the equation so that all $y$ terms are on one side and all $x$ terms are on the other.
• Steps to apply the method of separation of variables

  1️⃣ Separate Variables

  2️⃣ Integrate

  3️⃣ Solve for $y$

  4️⃣ Apply Initial Conditions
• What is the second step in applying separation of variables?

  Integrate both sides
• After integrating both sides in separation of variables, the next step is to solve for y
• What is the general solution to $\frac{dy}{dx} =$$\frac{x}{y}$?


  $y =$ \pm \sqrt{x^{2} + 2C}
• When separating variables, it is necessary to add a constant of integration on only one side.
• What is a differential equation?

  Equation relating function and derivatives
• Match the type of solution with its description:

  General Solution ↔️ Contains arbitrary constants

  Particular Solution ↔️ Satisfies initial conditions
• A particular solution is obtained from the general solution by using given initial conditions
• A differential equation is an equation that relates a function with its derivatives
• A general solution to a differential equation contains arbitrary constants.
• A general solution represents a single function, while a particular solution represents a family of functions.

  False
• What is a particular solution to a differential equation?

  Satisfies specific initial conditions
• A particular solution is a single function with no undefined constants
• Initial conditions are values given for the function or its derivatives at a specific point.
• Steps to apply the method of separation of variables

  1️⃣ Separate Variables

  2️⃣ Integrate

  3️⃣ Solve for $y$
• What is the general solution to the differential equation $\frac{dy}{dx} =$$2x$?


  $y =$$x^{2} +$$C$
• A particular solution is a solution to a differential equation that satisfies specific initial conditions
• What are initial conditions used for in differential equations?

  To find particular solutions
• Steps in the method of separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$

  4️⃣ Apply initial conditions
• When integrating both sides of a differential equation in separation of variables, you must add a constant of integration
• The final step in solving a differential equation using separation of variables is to solve for y
• Match the term with its definition:

  Particular Solution ↔️ Solution satisfying initial conditions

  Initial Conditions ↔️ Values at a specific point
• The general solution to \ln|y| = \frac{1}{2}x^{2} + C</latex> is $y =$$Ke^{\frac{1}{2}x^{2}}$, where $K$ is a replacement for e^{C}
• What is the value of $K$ in the particular solution $y =$$Ke^{\frac{1}{2}x^{2}}$ if $y(0) =$$2$?


  $2$
• When integrating both sides of a separated differential equation, you must add a constant of integration to only one side.
• The first step in applying separation of variables to $\frac{dy}{dx} =$$\frac{x}{y}$ is to separate the variables
• What is the result of separating variables in the equation \frac{dy}{dx} = \frac{x}{y}</latex>?

  $y \, dy =$$x \, dx$
• The general solution to $\frac{y^{2}}{2} =$$\frac{x^{2}}{2} +$$C$ is $y =$ \pm \sqrt{x^{2} + 2C}