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.