Question

if dy/dx = ysec^2(x) and y= 5 when x = 0, then y=?

А. e^tanx +4
B. e^tanx +5
C. 5e^tanx
D. tanx + 5
E. tanx + 5e^x

Answer 1

Answer:

C. $\displaystyle y = 5e^{tan(x)}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. 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
• Function Notation
• |Absolute Values|
• Anything to the 0th power is 1

Algebra II

• Logarithms and Natural logs
• Euler's number e

Calculus

Derivatives

Derivative Notation

Trig Derivatives

Differential Equations

• Separation of variables
• General and particular solutions

Antiderivatives - Integration

Integration Constant C

Trig Integration

Logarithmic Integration

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \frac{dy}{dx} = ysec^2(x)$

x = 0, y = 5

Step 2: Rewrite Differential

Separation of variables

1. [Division Property of Equality] Isolate y terms together:                             $\displaystyle \frac{1}{y}\frac{dy}{dx} = sec^2(x)$
2. [Multiplication Property of Equality] Isolate x terms together:                   $\displaystyle \frac{1}{y}dy = sec^2(x)dx$

Step 3: Find General Solution

1. [Equality Property] Integrate both sides:                                                     $\displaystyle \int {\frac{1}{y}} \, dy = \int {sec^2(x)} \, dx$
2. [1st Integral] Integrate [Logarithmic Integration]:                                         $\displaystyle ln|y| = \int {sec^2(x)} \, dx$
3. [2nd Integral] Integrate [Trig Integration]:                                                   $\displaystyle ln|y| = tan(x) + C$
4. [Equality Property] e both sides:                                                                   $\displaystyle e^{ln|y|} = e^{tan(x) + C}$
5. Simplify:                                                                                                         $\displaystyle |y| = Ce^{tan(x)}$
6. Rewrite:                                                                                                         $\displaystyle y = \pm Ce^{tan(x)}$

Step 4: Find Particular Solution

1. Substitute in variables [Function]:                                                               $\displaystyle |5| = Ce^{tan(0)}$
2. Evaluate absolute value:                                                                               $\displaystyle 5 = Ce^{tan(0)}$
3. Evaluate trig:                                                                                                 $\displaystyle 5 = Ce^0$
4. Evaluate exponent:                                                                                       $\displaystyle 5 = C(1)$
5. Multiply:                                                                                                             $\displaystyle 5 = C$
6. Rewrite:                                                                                                         $\displaystyle C =5$
7. Substitute in C [General Solution]:                                                               $\displaystyle y = 5e^{tan(x)}$

∴ Our answer is C.

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

Unit: Differential Equations

Book: College Calculus 10e