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

Consider the differential equation $\frac{d y}{d x} = x y + x$ . Which of the following best describes this equation?

It is neither linear nor separable.

It is both linear and separable.

It is separable but not linear.

It is linear but not separable.

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Verified step by step guidance

Step 1: Understand the definition of a linear differential equation. A first-order linear differential equation has the form:

dy/dx + P(x)y = Q(x)

, where P(x) and Q(x) are functions of x.

Step 2: Rewrite the given equation

dy/dx = xy + x

. Observe that it can be rearranged as

dy/dx - xy = x

, which matches the form of a linear differential equation with

P(x) = -x

and

Q(x) = x

Step 3: Determine if the equation is separable. A separable differential equation can be written in the form

dy/dx = g(x)h(y)

, where the variables x and y can be separated into two distinct functions. In this case,

dy/dx = xy + x

cannot be separated into such a form because the term

x

is added to

xy

, making separation impossible.

Step 4: Conclude that the equation is linear because it fits the form of a linear differential equation, but it is not separable because the variables cannot be separated into distinct functions of x and y.

Step 5: Verify the classification of the equation. Based on the analysis, the correct description is: 'It is linear but not separable.'