Differential Equations and Logarithms

The video tutorial explains how to solve a differential equation by checking for exactness and integrating to find the potential function. It demonstrates the process of verifying exactness by comparing partial derivatives and then integrating to find the general solution. Finally, it shows how to solve the equation for y using logarithmic properties.

What is the form of the given differential equation?

• Homogeneous

• Exact

• Linear

• Non-linear

What must be true for a differential equation to be exact?

• M and N are constants

• M equals N

• Partial of M with respect to y equals partial of N with respect to x

• Partial of M with respect to x equals partial of N with respect to y

What are the functions M and N in the rewritten differential form?

• M = x, N = y

• M = e^x, N = e^y

• M = e^y, N = e^x

• M = y, N = x

What is the result of integrating e^x with respect to x?

• ln(x) + C

• x^2 + C

• e^x + C

• e^x + a(y)

What is the function a(y) determined to be?

• e^x

• e^y

• ln(y)

• x + y

What is the general solution of the differential equation?

• x + y = C

• ln(x) + ln(y) = C

• e^x + e^y = 0

• e^x + e^y = C

How do you solve for y in the general solution?

• Multiply both sides by e^x

• Add e^x to both sides

• Subtract e^x from both sides

• Subtract e^y from both sides

What operation is used to isolate y in the equation e^y = C - e^x?

• Multiplication

• Division

• Logarithm

• Exponentiation

What is the final expression for y in terms of C and x?

• y = C - e^x

• y = e^x - C

• y = ln(C - e^x)

• y = ln(C + e^x)

What property of logarithms is used to solve for y?

• Quotient property

• Change of base property

• Product property

• Power property