Question

Find the particular solution of the differential equation?
/=5^3+9^2, when =1, =8

Answer 1

`\displaystyle s = (5t^4)/(4) + (9)/(t) - (9)/(4)`

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

• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

Calculus

Derivatives

Derivative Notation

Solving Differentials - Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Explanation:

*Note:

Ignore the Integration Constant C on the left hand side of the differential equation when integrating.

Step 1: Define

`\displaystyle (ds)/(dt) = 5t^3 + (9)/(t^2)`

t = 1

s = 8

Step 2: Integrate

1. [Derivative] Rewrite [Leibniz's Notation]:
   `\displaystyle ds = (5t^3 + (9)/(t^2))dt`
2. [Equality Property] Integrate both sides:
   `\displaystyle \int {} \, ds = \int {(5t^3 + (9)/(t^2))} \, dt`
3. [Left Integral] Reverse Power Rule:
   `\displaystyle s = \int {(5t^3 + (9)/(t^2))} \, dt`
4. [Right Integral] Rewrite [Integration Property - Addition]:
   `\displaystyle s = \int {5t^3} \, dt + \int {(9)/(t^2)} \, dt`
5. [Right Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle s = 5\int {t^3} \, dt + 9\int {(1)/(t^2)} \, dt`
6. [Right Integrals] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle s = 5\int {t^3} \, dt + 9\int {t^(-2)} \, dt`
7. [Right Integrals] Reverse Power Rule:
   `\displaystyle s = 5((t^4)/(4)) + 9((t^(-1))/(-1)) + C`
8. [Right Integrals] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle s = 5((t^4)/(4)) + 9((1)/(t)) + C`
9. Multiply:
   `\displaystyle s = (5t^4)/(4) + (9)/(t) + C`

Step 3: Solve

1. Substitute in variables:
   `\displaystyle 8 = (5(1)^4)/(4) + (9)/(1) + C`
2. Evaluate exponents:
   `\displaystyle 8 = (5)/(4) + (9)/(1) + C`
3. Divide:
   `\displaystyle 8 = (5)/(4) + 9 + C`
4. Add:
   `\displaystyle 8 = (41)/(4) + C`
5. [Subtraction Property of Equality] Isolate C:
   `\displaystyle (-9)/(4) = C`
6. Rewrite:
   `\displaystyle C = (-9)/(4)`

Particular Solution:
`\displaystyle s = (5t^4)/(4) + (9)/(t) - (9)/(4)`

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

Unit: Differentials Equations and Slope Fields

Book: College Calculus 10e