Advanced Math Solutions – Ordinary Differential Equations Calculator, Bernoulli ODE

Last post, we learned about separable differential equations. In this post, we will learn about Bernoulli differential equation, which will require us to refresh our brains on linear first order differential equations. A Bernoulli differential equation is a differential equation that is written in the form:

y^'+p(x)y=q(x)y^n

where p(x) and q(x) are continuous functions on a given interval and n is a rational number. The concept of Bernoulli differential equations is to make a nonlinear differential equation into a linear differential equation. If n=0 or n=1, then the equation is linear. Bernoulli’s equation is used, when n is not equal to 0 or 1.


How to solve Bernoulli differential equations:

1. Put the differential equation in the form of Bernoulli’s equation
2. Divide by y^n
3. Put the equation in the form \frac{1}{1-n}v^'+p(x)v=q(x)
• We do this because we set v=y^{1-n} and v^'=(1-n)y^{-n}y^'
4. Solve the linear first order differential equation
• We learned how to do this in a previous post

We will solve the first example step by step (click here):

1. Put the differential equation in the form of Bernoulli’s equation

2. Divide by y^n

3. Put the equation in the form \frac{1}{1-n}v^'+p(x)v=q(x)

4. Solve linear first order differential equation

Here’s another example (click here):

As long as you memorize Bernoulli’s equation, the equation in Step 3, and how to solve linear first order differential equations, Bernoulli differential equations should be a piece of cake, even if they do take a long time to solve.

Fun fact: Jacob Bernoulli, who found Bernoulli’s equation, is brothers with Johann Bernoulli, who supposedly discovered L’Hopital’s rule. The Bernoulli family had many brilliant mathematicians.

Until next time,

Leah

Advanced Math Solutions – Ordinary Differential Equations Calculator, Separable ODE

Last post, we talked about linear first order differential equations. In this post, we will talk about separable differential equations. A separable differential equation is a nonlinear first order differential equation that can be written in the form:

N(y)\frac{dy}{dx}=M(x)

A separable differential equation is separable if the variables can be separated. Separable differential equations are pretty simple and do not require many steps to solve.

How to solve separable differential equations:

1. Rewrite the differential equation as N(y)dy=M(x)dx
2. Integrate both sides
• Don’t forget to add the unknown constant
3. Solve for y(x)

We will solve the first example step by step (click here):

\frac{dy}{dx}=\frac{3x+1}{4y}

1. Rewrite the differential equation

2. Integrate both sides

3. Solve for y(x)

Here’s another example (click here):

Last example (click here):

I think separable differential equations are the easiest ordinary differential equations. There are very few steps to solve them and they are easy to remember. Memorize the steps and you’ll be good to go!

Until next time,

Leah

Advanced Math Solutions – Ordinary Differential Equations Calculator, Linear ODE

Ordinary differential equations can be a little tricky. In a previous post, we talked about a brief overview of ODEs. In this post, we will focus on a specific type of ODE, linear first order differential equations. A linear first order differential equation is an ODE that can be put in the form

\frac{dy}{dx}+p(x)y=q(x)

where p and q are continuous functions on a given interval.  In linear first order differential equations, we can derive a formula for the solution above.


How to solve a linear first order differential equation:

1. Put the equation into form \frac{dy}{dx}+p(x)y=q(x)
• This step is very important. You cannot solve, unless the equation is in this form.
2. Set p(x)=(\ln(μ(x)))' and solve for μ(x)
• μ(x) is known as the integrating factor
• Integrate both sides
• Don’t forget the unknown constant, c, and when you can combine multiple unknown constants
3. Multiply the equation in Step 1 by μ(x) and simplify
4. Use the product rule, f\cdot g^'+f^'\cdot g on the left side of the equation to make the left side of the equation equal to (μ(x)y(x))' and write it as that:
   
   μ(x)\frac{dy}{dx}+μ(x)p(x)y=μ(x)q(x)\quad\rightarrow\quad(μ(x)y(x))^'=μ(x)q(x)
5. Integrate and solve for y(x)
• Don’t forget the unknown constant and when you can combine multiple unknown 

We will go through this first example step by step (click here for a more detailed step by step): 

1. Put the equation in the form of a first order linear ODE

2. Solve for µ(x)

We integrate both sides to find µ(x).

3. Multiply by µ(x) and simplify

4. Make the left side equal to the product rule

5. Integrate and solve for y(x)

Here’s another example (click here):

Last example (click here):

Linear first order differential equations may look intimidating at first sight. These problems do take some time to solve, but are not very hard. The most important thing to remember is to not forget about the unknown constant, c.

Until next time,

Leah