# Riccati differential equations

• mcmaster1987
In summary, Riccati differential equations are a type of equation that deals with rates of change. When working with this equation, it is important to keep in mind the differential coefficient and the differential relation. The differential coefficient is used to calculate the rate of change and the differential relation is used to calculate the direction of the change. When solving for the general solution of this equation, it is important to remember to use the substitution method and to take R=-1.
mcmaster1987
riccati differential equations

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how to find general solution of this question

du/dt=u^2+t^2

i work hard but i do nat know this form of riccati equation. i know when special solution is given however there is no special soltion such that u=u1(t) in this question.

When I make the usual substitution:

$$u=\frac{y'}{Ry}$$

with $R=-1$ in your case (see info about Riccati equation),

I get:

$$y''+t^4y=0$$

Suppose you had to come up with an analytic expression for the eqn. in y. What would you do?

jackmell said:
When I make the usual substitution:

$$u=\frac{y'}{Ry}$$

with $R=-1$ in your case (see info about Riccati equation),

I get:

$$y''+t^4y=0$$

Suppose you had to come up with an analytic expression for the eqn. in y. What would you do?

why do you make u=y'/Ry and why you take R=-1

and i think, you have made an error process.

because you found an equation y''+yt^4=0

but i found y''+yt^2=0. and then i used second order linear differential equation thecniques.

After that i found y=C1e^it+C2e^(-it). i think this is not true.

Thank you for your interest my question.

Ok, my bad. It should be as you said and that's called the parabolic cylinder differential equation:

$$y''+x^2y=0$$

But that's not solved using ordinary techniques. You could however, use power series and that's what I was referring to above. Say you get it in the form:

$$y(x)=\sum_{n=0}^{\infty}a_nx^n$$

Then the solution to the original DE is:

$$u(x)=-\frac{\frac{d}{dx} \sum_{n=0}^{\infty}a_nx^n}{\sum_{n=0}^{\infty}a_nx^n}$$

Nothing wrong with that is there?

## 1. What is a Riccati differential equation?

A Riccati differential equation is a type of non-linear differential equation that has the form dy/dx = f(x) + g(x)y + h(x)y2, where f(x), g(x), and h(x) are functions of the independent variable x.

## 2. What is the history behind Riccati differential equations?

Riccati differential equations were first studied by Italian mathematician Jacopo Francesco Riccati in the 18th century. They were originally used to model problems in physics and engineering, but have since been applied to a wide range of fields such as economics and control theory.

## 3. What are some applications of Riccati differential equations?

Riccati differential equations have many applications in physics, engineering, and other fields. They are commonly used to model problems involving growth or decay, as well as in control theory to design optimal control systems. They are also used in economics to study optimal economic growth and in chemistry to model chemical reactions.

## 4. How are Riccati differential equations solved?

Riccati differential equations cannot be solved using traditional methods such as separation of variables or the method of undetermined coefficients. However, there are several techniques for solving them, including the substitution method, the Lie algebraic method, and the linearization method. In some cases, numerical methods may also be used to approximate solutions.

## 5. What are some challenges with solving Riccati differential equations?

One of the main challenges with solving Riccati differential equations is that they are non-linear, which means that traditional methods for solving differential equations cannot be applied. In addition, there is no general solution for all types of Riccati differential equations, so different techniques may need to be used for different types of problems. Furthermore, it can be difficult to find exact solutions, so numerical methods may be necessary for some problems.

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