Differential Equations: Solutions (Level 3 of 4)

This video continues the exploration of verifying solutions to ordinary differential equations (ODEs) and determining appropriate intervals of definition. It includes three examples: verifying a trigonometric function as a solution, using the second derivative for verification, and applying the fundamental theorem of calculus. The video emphasizes the importance of understanding derivatives, trigonometric identities, and the domain of functions. It concludes with a preview of the next video on partial differential equations.

What is the first step in verifying that y = 5 * tan(5x) is a solution to the differential equation y' = 25 + y^2?

• Graph the function.

• Find the first derivative of the function.

• Substitute the function into the differential equation.

• Find the second derivative of the function.

Which trigonometric identity is used to simplify the expression in the verification of y = 5 * tan(5x)?

• sin^2(x) + cos^2(x) = 1

• 1 + tan^2(x) = sec^2(x)

• tan(x) = sin(x)/cos(x)

• cos(2x) = cos^2(x) - sin^2(x)

For the function y = cos(t) * ln(cos(t)) + t * sin(t), what must be true about the argument of the logarithm for the function to be defined?

• The argument must be less than 0.

• The argument must be greater than 0.

• The argument must be equal to 0.

• The argument can be any real number.

In which quadrants of the unit circle is the function y = cos(t) * ln(cos(t)) + t * sin(t) defined?

• Quadrants I and II

• Quadrants II and III

• Quadrants III and IV

• Quadrants I and IV

What mathematical theorem is invoked to find the derivative of the function y = e^(t^2) * integral from 0 to t of e^(-s^2) ds + e^(t^2)?

• Pythagorean Theorem

• Mean Value Theorem

• Fundamental Theorem of Calculus

• Intermediate Value Theorem

What is the interval of definition for the function y = e^(t^2) * integral from 0 to t of e^(-s^2) ds + e^(t^2)?

• From 0 to infinity

• From negative infinity to 0

• From negative infinity to positive infinity

• From 0 to 1

What happens to the integral term when simplifying the expression for the function y = e^(t^2) * integral from 0 to t of e^(-s^2) ds + e^(t^2)?

• It doubles.

• It becomes zero.

• It cancels out.

• It remains unchanged.