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

Find the general solution of the differential equation $y^{4} + y^{3} + y^{2} = 0$ .

y (x) = C_{1} + C_{2} x + C_{3} e^{i x} + C_{4} e^{- i x}

y (x) = C_{1} + C_{2} x + C_{3} e^{\frac{- 1 + i \sqrt{3}}{2} x} + C_{4} e^{\frac{- 1 - i \sqrt{3}}{2} x}

y (x) = C_{1} + C_{2} x + C_{3} e^{\frac{- 1 + i \sqrt{2}}{2} x} + C_{4} e^{\frac{- 1 - i \sqrt{2}}{2} x}

y (x) = C_{1} + C_{2} x + C_{3} e^{- x} + C_{4} e^{0 x}

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Verified step by step guidance

Step 1: Start by analyzing the given differential equation y^{(4)} + y''' + y'' = 0. This is a linear homogeneous differential equation with constant coefficients.

Step 2: Assume a solution of the form y(x) = e^{rx}, where r is a constant. Substitute y(x) = e^{rx} into the differential equation to obtain the characteristic equation.

Step 3: The characteristic equation is derived by substituting y(x) = e^{rx} into the differential equation, resulting in r^4 + r^3 + r^2 = 0. Factorize this equation to find the roots.

Step 4: Factorize r^4 + r^3 + r^2 = 0 as r^2(r^2 + r + 1) = 0. This gives two types of roots: r = 0 (with multiplicity 2) and the roots of r^2 + r + 1 = 0, which are complex roots.

Step 5: Solve r^2 + r + 1 = 0 using the quadratic formula r = (-b ± √(b^2 - 4ac)) / 2a. Here, a = 1, b = 1, and c = 1. The roots are complex numbers, and the general solution is formed by combining the real and complex roots into the form y(x) = C_1 + C_2 x + C_3 e^{r_1 x} + C_4 e^{r_2 x}, where r_1 and r_2 are the complex roots.