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

Solve the differential equation $4 x^{2} y^{''} + 9 x y^{'} + y = x^{2} - x$ using the method of variation of parameters. Which of the following is the general solution?

y = C__{1} x^{- \frac{1}{4}} + C__{2} x^{- 1} + \frac{1}{12} x^{2} - \frac{1}{20} x

y = C__{1} e^{x} + C__{2} e^{- x} + x^{2} - x

y = C__{1} x^{2} + C__{2} x + \frac{1}{4} x^{2} - \frac{1}{8} x

y = C__{1} x^{\frac{1}{2}} + C__{2} x^{- \frac{1}{2}} + \frac{1}{8} x^{2} + \frac{1}{10} x

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Verified step by step guidance

Step 1: Rewrite the given differential equation in standard form. Divide through by 4x^2 to normalize the coefficient of y'' to 1. The equation becomes y'' + (9/(4x))y' + (1/(4x^2))y = (1/4) - (1/(4x)).

Step 2: Solve the associated homogeneous equation y'' + (9/(4x))y' + (1/(4x^2))y = 0. Use the method of trial solutions, assuming y = x^r, and substitute into the homogeneous equation to find the characteristic equation for r.

Step 3: Solve the characteristic equation obtained in Step 2 to find the roots r1 and r2. These roots will give the complementary solution y_c = C_1 x^r1 + C_2 x^r2.

Step 4: Apply the method of variation of parameters to find a particular solution y_p for the non-homogeneous equation. Use the formula y_p = u1(x)y1 + u2(x)y2, where y1 and y2 are the solutions from the homogeneous equation, and u1(x) and u2(x) are functions determined by solving a system of equations derived from the original differential equation.

Step 5: Combine the complementary solution y_c and the particular solution y_p to form the general solution y = y_c + y_p. Simplify the expression to match one of the given options.