Differential Equations: Solutions (Level 4 of 4)

This video tutorial covers the verification of solutions to Partial Differential Equations (PDEs). It begins with an introduction to PDEs and their importance, especially for those who need a calculus refresher. The video then provides three examples of verifying PDE solutions, each with detailed steps on finding partial derivatives and ensuring both sides of the equation match. The examples include using natural logarithms, subscript notation, and exponential and sine functions. The tutorial concludes with a preview of the next topic on implicit solutions of Ordinary Differential Equations (ODEs).

What is the main focus of this video tutorial?

• Solving ordinary differential equations

• Verifying solutions to partial differential equations

• Learning about integral calculus

• Exploring algebraic equations

In the first example, what is the given solution to verify?

• u = ln(x^2 + y^2)

• u = e^(x+y)

• u = sin(x + y)

• u = x^2 + y^2

What is the purpose of finding partial derivatives in the first example?

• To solve the equation

• To verify the solution

• To find the interval of definition

• To simplify the equation

In the second example, what notation is used for the PDE?

• Vector notation

• Matrix notation

• Subscript notation

• Leibniz notation

What is the solution to verify in the second example?

• u = tan(x + at)

• u = ln(x - at)

• u = sin(x - at)

• u = cos(x + at)

In the final example, what is the form of the solution to verify?

• u = alpha^2 * t * x

• u = ln(alpha^2 * t) * tan(x)

• u = e^(-alpha^2 * t) * sin(x)

• u = e^(alpha^2 * t) * cos(x)

What is the key step after finding all partial derivatives in the final example?

• Graphing the solution

• Finding the integral of the solution

• Substituting them into the PDE

• Assigning an interval of definition