Differential Equations and Integrals

The video tutorial provides practice in finding general solutions to separable differential equations. It begins with an introduction to the concept, explaining how to separate variables and treat differentials like variables. The instructor then solves two examples: the first involves the equation dy/dx = e^x/y, and the second involves dy/dx = y^2 * sin(x). Each example is solved step-by-step, demonstrating the integration process and the application of the reverse power rule. The video concludes with the general solutions for both examples.

What is the first step in solving a separable differential equation?

• Separate the variables

• Integrate both sides immediately

• Multiply by a constant

• Differentiate both sides

In the equation dy/dx = e^x/y, what should you multiply both sides by to start separating variables?

• y

• dx

• e^x

• dy

What is the integral of y dy?

• y^2/2

• y^3/3

• 1/y

• ln(y)

What is the anti-derivative of e^x?

• e^x + C

• x^e + C

• ln(x) + C

• 1/e^x + C

In the second example, what is the form of the differential equation given?

• dy/dx = y * ln(x)

• dy/dx = x^2 * cos(y)

• dy/dx = e^x/y

• dy/dx = y^2 * sin(x)

What is the anti-derivative of y^(-2) dy?

• ln(y)

• y^2/2

• -1/y

• y^(-1)

What is the anti-derivative of sin(x) dx?

• -cos(x) + C

• cos(x) + C

• sin(x) + C

• -sin(x) + C

How can you express the solution y in terms of x for the second example?

• y = 1/(sin(x) + C)

• y = cos(x) + C

• y = 1/(cos(x) + C)

• y = sin(x) + C

What is the purpose of adding a constant C in the solution of a differential equation?

• To account for initial conditions

• To make the equation more complex

• To simplify the integration

• To eliminate variables

What is the general solution for a separable differential equation?

• A solution that does not involve integration

• A solution that only works for linear equations

• A solution that includes a constant of integration

• A specific solution for given initial conditions