Unusual Solution to Riccati Differential Equation with ln x Particular Solution

In summary, the conversation discusses a Riccatti Equation where the answer obtained after solving it is described as "weird." The person attempted to find a particular solution by substituting values and transforming it into a Bernoulli and linear equation, but encountered a complicated integrating factor and obtained a different solution than expected.
  • #1
Ptopenny
3
0

Homework Statement


This is a Riccatti Equation which my answer is very weird after i solved it..
dy/dx+2y^2=(y/x lnx)+2(ln x)^2 where y=ln x is a particular solution


Homework Equations




The Attempt at a Solution



i first let y=ln x+ 1/w
and dy/dx=1/x-1/w^2
and substitute y and dy/dx into it place and form a ricatti equation, transforming it into bernoulli and linear equation... but the integrating factor is very very long and complicated...
 
Physics news on Phys.org
  • #2
and the solution i get is also very weird y=(x^2/2)+ln x + 1/(2w) which is totally different from what i wanted....
 

What is a Ricatti Differential Equation?

A Ricatti Differential Equation is a type of non-linear ordinary differential equation that is used to model various physical phenomena in fields such as physics, engineering, and economics. It is named after the Italian mathematician Jacopo Riccati who first studied these equations in the 1700s.

What is the general form of a Ricatti Differential Equation?

The general form of a Ricatti Differential Equation is:
y' = a(x)y^2 + b(x)y + c(x)
where y is the dependent variable, x is the independent variable, and a(x), b(x), and c(x) are functions of x.

What are the key features of a Ricatti Differential Equation?

The key features of a Ricatti Differential Equation include:
1. It is a non-linear equation, meaning that the dependent variable appears multiple times and is raised to a power.
2. It is a first-order differential equation, meaning that it involves only first derivatives of the dependent variable.
3. It can be transformed into a linear differential equation by making a specific substitution.
4. It has a general solution that can be expressed in terms of an arbitrary constant.

What are some applications of Ricatti Differential Equations?

Some applications of Ricatti Differential Equations include:
1. Modeling population growth in biology and ecology.
2. Studying the stability of control systems in engineering.
3. Calculating the optimal control policy in economics.
4. Analyzing the behavior of certain physical systems, such as the motion of a pendulum.

How do you solve a Ricatti Differential Equation?

There is no general method for solving all types of Ricatti Differential Equations. However, there are some techniques that can be used for specific cases. These include:
1. Making a substitution to transform the equation into a linear differential equation.
2. Using an integrating factor to solve the equation.
3. Using numerical methods, such as Euler's method or Runge-Kutta methods.
4. In certain cases, the equation can be solved using special functions, such as the hypergeometric function.

Similar threads

  • Calculus and Beyond Homework Help
Replies
21
Views
754
  • Calculus and Beyond Homework Help
Replies
5
Views
191
  • Calculus and Beyond Homework Help
Replies
8
Views
708
  • Calculus and Beyond Homework Help
Replies
2
Views
669
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
169
  • Calculus and Beyond Homework Help
Replies
3
Views
861
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
893
Back
Top