Riccati Equation. Unsure what substitution to use

In summary, a person is struggling with linearizing a given equation and solving the resulting 2nd order linear equation. They are unsure what to substitute for y and have tried various methods such as the power series and Bessel function, but are unsure of the correct solution. Another person agrees with their result and explains how to return to a function in y.
  • #1
Trestal
3
0
Hi,
I have the equation
y' = y^2 + x^2 and am asked to linearise the equation with the appropriate substitution and then solve the resulting 2nd order linear equation.

My issue is I am unsure what to substitute in for y. I can't seem to find a choice for y which the differential will be a higher order so that I can make the aux equation to solve for some value c.
 
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  • #2
Hl!

Let y(x) = -(1/f(x))*(df/dx)
leads to (d²f/dx²)+x²f(x)=0
 
  • #3
once you have the second order linear ODE, how do you solve it and then get back to the original Riccati equation? I tried power series method but it looks messy and I'm not sure if it's right, I then tried expressing as a Bessel function and get f= c1 x^1/2 J 1/4 (1/2 x^2) + c2 x^1/2 J -1/4 (1/2 x^2), but then how to return to a function in y?
 
  • #4
Going back to the power series soln, I get a soln for f, then I differentiated f and then substituted it all back into y = -1 f'/f to get a soln for y. The soln for y has power series on numerator and denominator and have different powers of x thru it.
 
  • #5
Hi !
I agree with your result :
f= c1 x^1/2 J 1/4 (1/2 x^2) + c2 x^1/2 J -1/4 (1/2 x^2)
To return to a function in y, you have to derivate f(x) and bring it back into y = -f'/f
This will leads to a fraction in which the numerator and the denominator are the sum of Bessel fonctions. This is the simplest way to express the result on a closed form.
 

1. What is the Riccati Equation?

The Riccati Equation is a type of differential equation that is used in control theory and other areas of mathematics. It is named after its creator, the Italian mathematician Jacopo Riccati.

2. What is the purpose of the Riccati Equation?

The Riccati Equation is used to describe the dynamics of a system over time. It is particularly useful in studying systems that are subject to external forces or disturbances.

3. How is the Riccati Equation solved?

The Riccati Equation can be solved using a variety of techniques such as substitution, transformation, and numerical methods. The specific approach used will depend on the specific form of the equation and the desired outcome.

4. What is substitution and how is it used in solving the Riccati Equation?

Substitution is a technique used in solving differential equations where a new variable is introduced to replace the original variable. This can help simplify the equation and make it easier to solve. In the case of the Riccati Equation, substitution is often used to transform it into a linear or Bernoulli form, which can then be solved using standard methods.

5. How do I know which substitution to use when solving the Riccati Equation?

The choice of substitution to use when solving the Riccati Equation will depend on the specific form of the equation and the desired outcome. Some common substitutions include using a logarithmic or exponential function to transform the equation, or introducing a new variable such as u = y'/y to reduce the equation to a linear form. It is important to carefully consider the form of the equation and the desired outcome before choosing a substitution.

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