Tuesday, November 3, 2015

Advanced Math Solutions – Ordinary Differential Equations Calculator, Exact Differential Equations

In the previous posts, we have covered three types of ordinary differential equations, (ODE). We have now reached the last type of ODE. In this post, we will talk about exact differential equations.

What is an exact differential equation?


There must be a 0 on the right side of the equation and M(x,y)dx and N(x,y)dx must be separated by a +.

Steps to solve exact differential equations:
  1. Verify that \frac{∂M(x,y)}{∂y}=\frac{∂N(x,y)}{∂x}
    • Find M(x,y) and N(x,y)
  2. Integrate \int M(x,y)dx or \int N(x,y)dy
    • This will help us find Ψ(x,y)
  3. Replace c with ƞ(x) if you integrated N(x,y) with respect to y, or ƞ(y) if you integrated M(x,y) with respect to x
  4. Compute ƞ(x) or ƞ(y)
  5. Solve to get the implicit or explicit solution, depending on which is preferred
    • Don’t forget to substitute ƞ(x) or ƞ(y)


Exact differential equations can be tricky. We will solve the first example step by step to help you better understand how to solve exact differential equations.


First example (click here):

1. Verify that \frac{∂M(x,y)}{∂y}=\frac{∂N(x,y)}{∂x}
M(x,y)=2xy-9x^2
N(x,y)=2y+x^2+1

2. Integrate \int N(x,y)dy


3. Replace c with ƞ(x)



4. Compute ƞ(x)



We took the derivative with respect to x of Ψ(x,y), which is equal to y+x^2 y+y^2+ƞ(x). We then compared the derivative to M(x,y), the equation we didn’t integrate. We then integrated both sides to solve for ƞ(x).

5. Find the implicit equation



Here’s another example (click here):



Exact differential equations may look scary because of the odd looking symbols and multiple steps. If you double check your work, memorize the steps, and practice, you can definitely get this concept down. Don’t be afraid and dive in!

Until next time,

Leah

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