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: $x^{2} y^{'} + x (x + 2) y = e^{x}$ .

y = \frac{e^{x}}{x} + C

y = x^{2} e^{x} + C

y = \frac{1}{x^{2}} [\int x e^{x} d x + C]

y = e^{x} + C x^{2}

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Verified step by step guidance

Step 1: Recognize that the given differential equation is a first-order linear differential equation of the form x²y' + x(x + 2)y = e^x. Rewrite it in standard form by dividing through by x² to isolate y': y' + (x + 2)/x y = e^x/x².

Step 2: Identify the integrating factor for the equation. The integrating factor is derived from the coefficient of y in the standard form, which is (x + 2)/x. Compute the integrating factor as e^(∫(x + 2)/x dx). Break this integral into simpler parts: ∫(x + 2)/x dx = ∫1 dx + ∫2/x dx = x + 2ln|x|.

Step 3: Write the integrating factor as e^(x + 2ln|x|). Simplify this expression using properties of exponents: e^(x + 2ln|x|) = e^x * e^(ln|x²|) = e^x * x².

Step 4: Multiply through the differential equation by the integrating factor (e^x * x²) to make the left-hand side an exact derivative. The equation becomes: d/dx[(e^x * x² * y)] = e^x * x.

Step 5: Integrate both sides with respect to x. The left-hand side integrates to e^x * x² * y, and the right-hand side requires integration of e^x * x. After integrating, solve for y to find the general solution: y = (1/x²) [∫x e^x dx + C], where C is the constant of integration.