Which of the following is the solution to the differential equation dy/dx=2xy/x^2+1 whose graph contains the points (0,1)? A. y=ex^2
B. y=x^2+1
C. y=ln(x^2+1)
D. y=1+ln(x^2+1)
E. y=sqrt(1+2ln(x^2+1))

B. y = x² + 1

General Formulas and Concepts:

Symbols

e (Euler's number) ≈ 2.7182

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right  

Equality Properties

Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of Equality

Algebra I

Functions

Algebra II

Logarithms - ln and e

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Slope Fields

Solving Differentials - Integrals

Integration Constant C

U-Substitution

ln Integration:

Step-by-step explanation:

*Note:

When solving differential equations in slope fields, disregard the integration constant C for variable y.

Step 1: Define

Point (0, 1)

Step 2: Rewrite Differential

Rewrite Leibniz Notation using Separation of Variables.

[Separation of Variables] Isolate x's together:                                             [Separation of Variables] Isolate y's together:                                            

Step 3: Integrate Pt. 1

Solving general form of differential using integration.

[Differential] Integrate both sides:                                                                 [Left Integral] ln Integration:                                                                          

Step 4: Identify Variables

Set up u-substitution for right integral.

u = x² + 1

du = 2xdx

Step 5: Integrate Pt. 2

[Right Integral] U-Substitution:                                                                       [Right Integral] ln Integration:                                                                        Back-Substitute:                                                                                             [Equality Property] Raise e on both sides:                                                    Simplify:                                                                                                          

General Form:

Step 6: Solve Particular Solution

Since both sides have absolute value, assume that the particular solution will be positive.

Substitute in point [General Form]:                                                                [Particular] |Absolute Value| Evaluate exponents:                                        [Particular] Evaluate absolute values:                                                           [Particular] Rewrite:                                                                                        

Substituting integration constant C into the general form:

Particular Solution:

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Differentials and Slope Fields

Book: College Calculus 10e

The decimal expansion would be .27 repeated