Differential Equations Flashcards

Differential Equations

Any equation which contains derivatives of functions is considered a differential equation.

Real World Applications

• Uninhibited Growth
• Newton's Law of Cooling

Solving Differential Equations with Initial Conditions

Vocabulary

• First Order: A differential equation with only the first derivative.
• Second Order: A differential equation with the highest order derivative of the second derivative.
• Initial Condition: A y-value at an arbitrary x-value, usually known as x_0.
• General Solution: A solution to the differential equation with some unknown constants.
• Particular Solution: A fully defined solution to a differential equation.

How to Solve These Equations

Separable

When all the variables can be separated by the = sign, integrate with respect to x and y on each side.

Examples

\frac{dy}{dx} = y can be rewritten as \frac{dy}{y} = dx.
Integrating both sides gives \int \frac{dy}{y} = \int dx, which results in \ln|y| = x + c.
Then, e^{\ln|y|} = e^{x+c}, which simplifies to y = Ce^x.

Another example:

\frac{dy}{dx} = (-5y + 10)x

\frac{dy}{-5y + 10} = x dx

\frac{dy}{-5(y - 2)} = x dx

\int \frac{dy}{y-2} = \int -5x dx

\ln(y-2) = \frac{-5x^2}{2} + C

y-2 = Ce^{\frac{-5x^2}{2}}

y = Ce^{\frac{-5x^2}{2}} + 2

Another example:
\frac{dy}{dx} = \frac{y}{x}

\frac{dy}{y} = \frac{dx}{x}

\int \frac{dy}{y} = \int \frac{dx}{x}

\ln|y| = \ln|x| + c

e^{\ln|y|} = e^{\ln|x| + c}

y = C|x|

y = Cx, x \neq 0

Steps to Solve Separable Differential Equations

1. Separate the variables. Be sure that the differentials are in the numerator.
   • 1a. If you have a log on one side, put all constants on the other side.
2. Integrate. Do not forget the constant.
   • 2a. Logs and Absolute Value:
     • y only: you can skip (reason not provided).
     • x: use absolute value.
3. Solve for y. Be careful with the constant.
4. Check your work by differentiating.