7.6 Finding General Solutions Using Separation of Variables

• What is a differential equation?

  Equation relating function to derivative
• The general solution to a differential equation includes an arbitrary constant
• The method of separation of variables involves separating $x$ and $y$ on opposite sides of the equation.
• Steps to solve differential equations using separation of variables:

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$
• Solve the differential equation $\frac{dy}{dx} =$$x^{2}y$


  $y =$$Ke^{\frac{1}{3}x^{3}}$
• The general solution to a differential equation includes an arbitrary constant
• The arbitrary constant in a general solution makes it cover a family of functions.
• Match the term with its definition:

  General Solution ↔️ Function satisfying equation with arbitrary constant

  Arbitrary Constant ↔️ A constant that can take any value
• The separation of variables method involves isolating the dependent and independent variables on opposite sides of the equation
• Steps for separation of variables:

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$
• Solve the differential equation \frac{dy}{dx} = \frac{x}{y}</latex>

  $y =$ \pm \sqrt{x^{2} + 2C}
• A general solution of a differential equation includes an arbitrary constant
• An arbitrary constant in a general solution allows for a family of functions that satisfy the differential equation.
• Solve the differential equation $\frac{dy}{dx} =$$2x$


  y = x^{2} + C</latex>
• What is a general solution of a differential equation?

  A function with an arbitrary constant
• An arbitrary constant in a general solution can take any value.
• The general solution of $\frac{dy}{dx} =$$2x$ is y = x^{2} + C
• Steps to solve a differential equation using separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$
• What is the first step in solving $\frac{dy}{dx} =$$x^{2}y$ using separation of variables?


  Separate variables
• The general solution of $\frac{dy}{y} =$$x^{2} dx$ is y = Ke^{\frac{1}{3}x^{3}}
• The differential equation $\frac{dy}{dx} =$$\frac{x}{y}$ can be solved using separation of variables.
• Match the steps of separation of variables with their descriptions:

  Separate Variables ↔️ Isolate $y$ and $dy$ on one side and $x$ and $dx$ on the other

  Integrate Both Sides ↔️ Calculate the integral of each side of the equation

  Solve for $y$ ↔️ Express $y$ in terms of $x$ and the constant of integration
• What method is used to solve certain types of differential equations by isolating variables and integrating?

  Separation of variables
• Steps for the method of separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides
• A differential equation relates a function to its derivatives
• What variables should be on opposite sides when separating variables in a differential equation?

  $y$ and $x$
• After integrating, you must solve for y
• The method of separation of variables involves integrating both sides of an equation
• What is the first step in solving $\frac{dy}{dx} =$$x^{2}y$ using separation of variables?


  $\frac{dy}{y} =$$x^{2}dx$
• The general solution for \frac{dy}{dx} = x^{2}y</latex> is $y =$$Ke^{\frac{1}{3}x^{3}}$, where $K =$$e^{C}$ and $C$ is the constant of integration
• A general solution of a differential equation must include an arbitrary constant
• Steps for finding a general solution using separation of variables

  1️⃣ Separate variables

  2️⃣ Integrate both sides

  3️⃣ Solve for $y$
• A general solution of a differential equation includes an arbitrary constant
• What are the three steps involved in the method of separation of variables?

  Separate variables, integrate, solve for y
• Steps to solve a differential equation using separation of variables

  1️⃣ Rewrite the equation with $y$ and $dy$ on one side and $x$ and $dx$ on the other

  2️⃣ Integrate both sides of the separated equation

  3️⃣ Express $y$ as a function of $x$ and the constant of integration
• The arbitrary constant in a general solution allows for a family of functions that all satisfy the differential equation.
• What is the final step in the method of separation of variables?

  Solve for y</latex>
• A general solution of a differential equation satisfies the equation and includes an arbitrary constant
• Steps to solve a differential equation using separation of variables

  1️⃣ Separate the variables $y$ and $x$

  2️⃣ Integrate both sides with respect to their respective variables

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


  $y =$$x^{2} +$$C$