Separable differential equations are equations that can be separated so that one variable is on one side, and the other variable is on the other side. This allows you to integrate each side to find the solution to the differential equation.

Here is an example:

Example problem 1:

Example problem 2:

Like the example above, sometimes answers can be left as implicit solutions if an explicit solution is too hard to find.

Example problem 3:

Bibliography:

Holzner, Steven. “Sorting Out Separable First Order Differential Equations.” Differential Equations for Dummies. Hoboken, NJ:Wiley Publishing, 2008. 41-49. Print.

Khan, Salman. “Separable Differential Equations Introduction.” Khan Academy. 23 Sept. 2014. Web. 18 Nov 2015.

Khan, Salman. “Particular Solution to Differential Equation Example.” Khan Academy. 23 Sept. 2014. Web. 18 Nov 2015.

*Note: the reason why we get f(y) instead of C when we take the integral of the partial derivative of Ψ with respect to x is because it is a partial derivative. Taking the partial derivative of f(y) with respect to x would result in 0, so there may be a function y that ‘disappeared’ in the Ψ(x) equation.

Example Problem #2:

Example Problem #3:

Bibliography:

Holzner, Steven. “Exploring Exact First Order Differential Equations and Euler’s Method.” Differential Equations for Dummies. Hoboken, NJ:Wiley Publishing, 2008. 63-70. Print.

Khan, Salman. “Exact Equations Example 1.” Khan Academy. 30 Aug. 2008. Web. 6 Nov 2015.

Khan, Salman. “Exact Equations Example 2.” Khan Academy. 30 Aug. 2008. Web. 7 Nov 2015.

Khan, Salman. “Exact Equations Example 3.” Khan Academy. 30 Aug. 2008. Web. 10 Nov 2015.

Tenenbaum, Morris, and Harry Pollard. “Lesson 9 – Exact Differential Equations.” Ordinary Differential Equations. New York: Dover Publications, 1963. 70-80. Print

If a linear first order differential equation is difficult to integrate, it can be converted to something that is easy to integrate. This is done by finding an integrating factor, which is a function μ(t). The integrating factor is then multiplied with the differential equation.

Here is an example to demonstrate how integrating factors work:

Practice Problem #1:

Bibliography:

Binder, Andrew. “Integrating Factor Method.” University of Minnesota, 17 Feb. 2012. PDF file. Web. 5 Oct. 2015.

Holzner, Steven. “Looking at Linear First Order Differential Equations.” Differential Equations for Dummies. Hoboken, NJ:Wiley Publishing, 2008. 26-29. Print.

In order to draw a slope field, you must plug in values for x and y into the differential equation. For example, if we plugged in (1,0) into the equation dy/dx=x^2, we would get dy/dx=1. Looking at the graph, we see that the slope at (1,0) is indeed 1. The graph also shows the same slope for equal x values. This makes sense because dy/dx is only affected by the x value.

Why is a slope field important?

Slope fields help us visualize the graph without actually solving it. The graph we draw is called an integral curve, where each point on the graph is tangent to the slope field. The specific solution of a differential equation depends on the point the graph contains. For example, the graph I drew of dy/dx = x^2 passes through the point (0,1).

Works Cited:

Holzner, Steven. “Welcome to the World of Differential Equations.” Differential Equations for Dummies. Hoboken, NJ:Wiley Publishing, 2008. 13-16. Print.

Khan, Salman. “Creating a Slope Field.” Khan Academy. 18 Sept. 2014. Web. 26 Sept 2015.

Martin, Mike. “First-Order ODE’s: Existence & Uniqueness, Slope Fields, & Qualitative Analysis.” Biomathdynamics.com. Johnson County Community College, 2012. Web. 24 Sept. 2015.