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 $e^{x} y \frac{d y}{d x} = e^{- y} + e^{- 3 x} - y$ by separation of variables.

y = - 3 x + C

y = \ln (e^{- 3 x} + C)

No solution by separation of variables; the equation is not separable.

y = - \ln (e^{- 3 x} + C)

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Verified step by step guidance

Step 1: Analyze the given differential equation: $ e^x y \frac{dy}{dx} = e^{-y} + e^{-3x} - y $. The goal is to determine if the equation can be solved using the method of separation of variables.

Step 2: Recall that separation of variables requires the equation to be expressible in the form $ f(y)dy = g(x)dx $, where the variables $ y $ and $ x $ can be separated completely.

Step 3: Attempt to rearrange the equation to isolate terms involving $ y $ and $ x $. Observe that $ e^x y \frac{dy}{dx} $ includes mixed terms of $ x $ and $ y $, and the right-hand side $ e^{-y} + e^{-3x} - y $ also mixes $ x $ and $ y $. This suggests the equation is not separable.

Step 4: Verify by substitution whether any proposed solutions (e.g., $ y = -3x + C $, $ y = \ln(e^{-3x} + C) $, or $ y = -\ln(e^{-3x} + C) $) satisfy the original equation. None of these solutions satisfy the equation when substituted back, confirming that separation of variables is not applicable.

Step 5: Conclude that the differential equation $ e^x y \frac{dy}{dx} = e^{-y} + e^{-3x} - y $ cannot be solved using the method of separation of variables due to the inherent mixing of $ x $ and $ y $ terms.