Integrating Factors for Solving Linear Differential Equations

In summary, the conversation discusses the process of dividing an equation by (1-x^2) to make it into a linear differential equation. However, there is difficulty in obtaining the value of Q(x). The conversation then goes on to talk about finding the integrating factor to continue with the process.
  • #1
mahmoud shaaban
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Can i have help with this linear differential equation ?
First, i divided by (1-x^2) to be like dy/dx + p(x)y= q(x). But i could not obtain Q(x).
Any help will be welcomed.
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  • #2
Let's look at what happens when we divide the RHS by $1-x^2$:

\(\displaystyle \frac{(1-x)\sqrt{1-x^2}}{1-x^2}=\frac{1-x}{\sqrt{1-x^2}}=\frac{1-x}{\sqrt{(1+x)(1-x)}}=\sqrt{\frac{1-x}{1+x}}\)

So, the ODE is now:

\(\displaystyle \d{y}{x}+\frac{x^2}{1-x^2}y=\sqrt{\frac{1-x}{1+x}}\)

Next, we want to compute the integrating factor:

\(\displaystyle \mu(x)=\exp\left(\int\frac{x^2}{1-x^2}\,dx\right)\)

Can you proceed?
 

FAQ: Integrating Factors for Solving Linear Differential Equations

What is a linear differential equation?

A linear differential equation is an equation that involves a function and its derivatives, where the function and its derivatives are only raised to the first power and are not multiplied together. It can be written in the form: y'(x) + p(x)y(x) = q(x), where y'(x) is the first derivative of y with respect to x, p(x) and q(x) are functions of x.

What are the solutions to a linear differential equation?

The solutions to a linear differential equation depend on the initial conditions given. For a first-order linear differential equation, there will be one solution that satisfies the given initial condition. For a higher-order linear differential equation, there will be multiple solutions that satisfy the given initial conditions.

How do I solve a linear differential equation?

To solve a linear differential equation, you can use various methods such as separation of variables, integrating factors, or finding the homogeneous and particular solutions. It is important to consider the initial conditions given when solving the equation.

What is the difference between a linear and a non-linear differential equation?

The main difference between a linear and a non-linear differential equation is that the variables in a linear differential equation are raised to the first power and are not multiplied together, while in a non-linear differential equation, the variables may be raised to higher powers or multiplied together. This makes solving linear differential equations easier compared to non-linear ones.

Where are linear differential equations used?

Linear differential equations have many applications in science, engineering, and mathematics. They are used to model various real-life situations, such as population growth, radioactive decay, and electrical circuits. They are also used in fields like physics, chemistry, and economics to study dynamic systems.

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