Differential Equations and Integrating Factors

This video tutorial explains how to use an integrating factor to convert a non-exact differential equation into an exact one. It begins by checking the exactness of the given equation using partial derivatives. When the equation is found to be non-exact, the tutorial demonstrates how to find an integrating factor that depends only on one variable. After applying the integrating factor, the equation becomes exact, allowing for the derivation of the general solution. The tutorial concludes with solving for the dependent variable and presenting the implicit form of the solution.

What is the condition for a differential equation to be considered exact?

• The partial derivative of M with respect to x equals the partial derivative of N with respect to y.

• The partial derivative of M with respect to y equals the partial derivative of N with respect to x.

• The partial derivative of N with respect to x equals zero.

• The partial derivative of M with respect to y equals zero.

What is the partial derivative of M with respect to y for the given equation?

• Two y to the power of negative three cosine X

• Negative y to the power of negative two

• Y to the power of negative two

• Negative two y to the power of negative three cosine X minus y to the power of negative two

Why is the given differential equation not exact?

• Because the partial of M with respect to y equals the partial of N with respect to x.

• Because the equation is homogeneous.

• Because the equation is linear.

• Because the partial of M with respect to y does not equal the partial of N with respect to x.

What is the purpose of finding an integrating factor?

• To make a non-linear equation linear.

• To simplify the equation by eliminating variables.

• To convert a non-exact differential equation into an exact one.

• To find the roots of the equation.

What is the integrating factor U(y) derived in the video?

• y to the power of negative two

• e to the power of two natural log absolute value of y

• e to the power of negative two natural log absolute value of y

• y squared

After applying the integrating factor, what is the new form of the differential equation?

• Cosine X plus y squared times DX plus x d y equals zero

• Cosine X plus y times DX plus x d y equals zero

• Cosine X times DX plus y d y equals zero

• X times DX plus y d y equals zero

What is the general solution of the exact differential equation after applying the integrating factor?

• Sine X minus x y equals a constant

• Cosine X minus x y equals a constant

• Cosine X plus x y equals a constant

• Sine X plus x y equals a constant

How do you solve for y in the general solution?

• Subtract cosine X from both sides and divide by y

• Add cosine X to both sides and multiply by y

• Subtract sine X from both sides and divide by x

• Add sine X to both sides and multiply by x

What is the explicit form of the general solution for y?

• y equals the quantity C minus sine X divided by X

• y equals the quantity C plus sine X divided by X

• y equals the quantity C minus cosine X divided by X

• y equals the quantity C plus cosine X divided by X

What is the role of the constant C in the general solution?

• It is an arbitrary constant representing the family of solutions.

• It represents a specific solution.

• It is the derivative of the solution.

• It is the initial condition of the solution.