Question

Find the solution of the differential equation f' (t) = t^4+91-3/t
having the boundary condition f(1) =1/4

Answer 1

Answer:

$\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}$

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

Calculus

Derivatives

Derivative Notation

Antiderivatives - Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle f'(t) = t^4 + 91 - \frac{3}{t}$

$\displaystyle f(1) = \frac{1}{4}$

Step 2: Integration

Integrate the derivative to find function.

1. [Derivative] Integrate:                                                                                   $\displaystyle \int {f'(t)} \, dt = \int {t^4 + 91 - \frac{3}{t}} \, dt$
2. Simplify:                                                                                                         $\displaystyle f(t) = \int {t^4 + 91 - \frac{3}{t}} \, dt$
3. Rewrite [Integration Property - Addition/Subtraction]:                               $\displaystyle f(t) = \int {t^4} \, dt + \int {91} \, dt - \int {\frac{3}{t}} \, dt$
4. [1st Integral] Integrate [Integral Rule - Reverse Power Rule]:                     $\displaystyle f(t) = \frac{t^5}{5} + \int {91} \, dt - \int {\frac{3}{t}} \, dt$
5. [2nd Integral] Integrate [Integral Rule - Reverse Power Rule]:                   $\displaystyle f(t) = \frac{t^5}{5} + 91t - \int {\frac{3}{t}} \, dt$
6. [3rd Integral] Rewrite [Integral Property - Multiplied Constant]:                 $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3\int {\frac{1}{t}} \, dt$
7. [3rd Integral] Integrate:                                                                                 $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C$

Our general solution is  $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C$.

Step 3: Find Particular Solution

Find Integration Constant C for function using given condition.

1. Substitute in condition [Function]:                                                               $\displaystyle f(1) = \frac{1^5}{5} + 91(1) - 3ln|1| + C$
2. Substitute in function value:                                                                         $\displaystyle \frac{1}{4} = \frac{1^5}{5} + 91(1) - 3ln|1| + C$
3. Evaluate exponents:                                                                                     $\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3ln|1| + C$
4. Evaluate natural log:                                                                                     $\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3(0) + C$
5. Multiply:                                                                                                         $\displaystyle \frac{1}{4} = \frac{1}{5} + 91 - 0 + C$
6. Add:                                                                                                               $\displaystyle \frac{1}{4} = \frac{456}{5} - 0 + C$
7. Simplify:                                                                                                         $\displaystyle \frac{1}{4} = \frac{456}{5} + C$
8. [Subtraction Property of Equality] Isolate C:                                               $\displaystyle -\frac{1819}{20} = C$
9. Rewrite:                                                                                                         $\displaystyle C = -\frac{1819}{20}$
10. Substitute in C [Function]:                                                                             $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}$

∴ Our particular solution to the differential equation is  $\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}$.

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

Unit: Integration

Book: College Calculus 10e