Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

Of the following, which is not a solution to the differential equation $y^{″} + 9 y = 0$ ?

y = 2 \sin (3 x)

y = 5 \cos (3 x)

y = - 4 \sin (3 x) + 7 \cos (3 x)

y = e^{3 x}

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Verified step by step guidance

Step 1: Understand the differential equation y'' + 9y = 0. This is a second-order linear homogeneous differential equation. The general solution to such equations is typically a combination of sine and cosine functions when the characteristic equation has real roots.

Step 2: Verify the given solutions by substituting each one into the differential equation. For example, substitute y = 2 sin(3x) into the equation. Compute y' and y'' using the derivative rules for trigonometric functions.

Step 3: Substitute y = 5 cos(3x) into the differential equation. Similarly, compute y' and y'' for this function and check if the equation y'' + 9y = 0 holds true.

Step 4: Substitute y = -4 sin(3x) + 7 cos(3x) into the differential equation. Compute y' and y'' for this combination of functions and verify if it satisfies y'' + 9y = 0.

Step 5: Finally, substitute y = e^(3x) into the differential equation. Compute y' and y'' for this exponential function and check if it satisfies y'' + 9y = 0. You will find that it does not satisfy the equation, as the exponential function does not align with the characteristic equation's solutions.