Notes on Differential Equations, Riemann Sums, and the Fundamental Theorem of Calculus

I. Differential Equations

  1. General Solutions

To find the general solution for the first-order differential equation:
\frac{dy}{dt} = y^3 t^2

  • Step 1: Separate variables by rewriting as ( \frac{1}{y^3} dy = t^2 dt ).

  • Step 2: Integrate both sides. For (\int \frac{1}{y^3} dy = -\frac{1}{2y^2} ) and (\int t^2 dt = \frac{t^3}{3} + C ).

  • Step 3: Combine results: (-\frac{1}{2y^2} = \frac{t^3}{3} + C) and solve for (y).

Tricks to remember:

  • Separate variables: This means you can keep all (y)s on one side and all (t)s on the other.

  • Integrate carefully: Always check your basic integration rules!

For the second-order differential equation:
\frac{d^2y}{dx^2} = \cos(5x)

  • Step 1: Integrate once to find the first derivative: (\frac{dy}{dx} = \int \cos(5x) dx = \frac{1}{5} \sin(5x) + C_1).

  • Step 2: Integrate again to find (y): (y(x) = \int \left(\frac{1}{5} \sin(5x) + C1\right) dx = -\frac{1}{25} \cos(5x) + C1x + C_2).

  1. Particular Solutions

To find the particular solution that passes through the point (1, 2) for:
\frac{dy}{dx} = \frac{x}{x^2 - 1}

  • Step 1: Integrate both sides as before: (y = \int \frac{x}{x^2 - 1} dx).

  • Step 2: Use substitution or partial fractions to solve the integral.

  • Step 3: Find (y) using the point (1, 2): Plug in to find any constants from integration.

Tricks to remember:

  • Plug in values: Always remember to use the given point to find constants.

For:
\frac{d^2y}{dx^2} = x^2 - 2, \quad \frac{dy}{dx} = -2

  • Do two integrations and then apply initial conditions similar to above.

  1. Applications of Differential Equations

For the acceleration due to gravity on planet ZibZab:
a(t) = -4 \ m/s^2

  • Step 1: Use (a(t) = \frac{d^2y}{dt^2}) to integrate twice to find position (y).

  • Step 2: Consider initial velocity and height when solving.

For the mouse running in a tube:
a(t) = \frac{1}{3} \cos(4t)

  • Follow similar steps as above to find velocity and position by integrating acceleration.

II. Riemann Sums

  1. Finding Riemann Sums

Given the function on the interval [0, 2]:
f(x) = x^2
To find the Left Riemann Sum for four subintervals, denoted as L4:

  • Step 1: Divide the interval into 4 parts: ([0,0.5], [0.5,1], [1,1.5], [1.5,2]).

  • Step 2: Calculate the height using the left endpoint of each subinterval: (f(0), f(0.5), f(1), f(1.5)).

  • Step 3: Multiply each height by the width of the interval and sum.

For Right Riemann Sum R4, do the same, but use the right endpoints.

For the Trapezoidal Riemann Sum T4, average left and right sums for the final estimate.

III. The Fundamental Theorem of Calculus

  1. Definite Integrals

Use definite integrals to calculate exact area:
Compute the integral:
\int_{0}^{2} (x + 1)(x - 3) \, dx

  • Step 1: Expand the function before integrating for easier calculation.

  • Step 2: Integrate and then evaluate at the limits 0 and 2.

Tricks to remember:

  • Always keep track of upper and lower limits before solving.

Compute:
\int_{0}^{\pi} \cos^2(\theta) \sin(\theta) \, d\theta

  • Use substitution for (\cos^2(\theta)) if you're comfortable with integration methods.

Compute:
\int_{1}^{5} \frac{x}{x - 1} \, dx

  • Remember to simplify before integrating and pay attention to potential limits of integration sometimes causing asymptotes.