# Calculate differential equation

• Chromosom
No, there is no restriction on the value of C. The general solution for this differential equation is y=\frac{x^2}{4}+C, where C is any constant.
Chromosom

## Homework Statement

$$y=xy^\prime-\left(y^\prime\right)^2$$

## The Attempt at a Solution

Unfortunately, I do not have any good idea. I tried $$y=xt(x)$$, but the equation only became worse.

Chromosom said:

## Homework Statement

$$y=xy^\prime-\left(y^\prime\right)^2$$

## The Attempt at a Solution

Unfortunately, I do not have any good idea. I tried $$y=xt(x)$$, but the equation only became worse.

Differentiate both sides with respect to $x$. You do have to check whether all the solutions of the resulting 2nd order ODE are solutions of this ODE.

1 person
Very clever :)

$$y^\prime=y^\prime+xy^{\prime\prime}-2y^\prime y^{\prime\prime}$$

$$xy^{\prime\prime}-2y^\prime y^{\prime\prime}=0$$

$$y^{\prime\prime}\left(x-2y^\prime\right)=0$$

Now we could expect that $$y=ax+b$$, but:

$$y^\prime=a$$

$$ax+b=ax-a^2$$

$$b+a^2=0$$

The last equation must be satisfied in order to be a solution. And of course second solution: $$y=\frac{x^2}{4}+C$$

Chromosom said:
Very clever :)

$$y^\prime=y^\prime+xy^{\prime\prime}-2y^\prime y^{\prime\prime}$$

$$xy^{\prime\prime}-2y^\prime y^{\prime\prime}=0$$

$$y^{\prime\prime}\left(x-2y^\prime\right)=0$$

Now we could expect that $$y=ax+b$$, but:

$$y^\prime=a$$

$$ax+b=ax-a^2$$

$$b+a^2=0$$

The last equation must be satisfied in order to be a solution.

This is correct.

And of course second solution: $$y=\frac{x^2}{4}+C$$

Is there any restriction on the value of $C$?

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical and natural phenomena in fields such as physics, engineering, and biology.

## 2. How do you solve a differential equation?

There are various methods to solve a differential equation, such as separation of variables, substitution, and integrating factors. The choice of method depends on the type of differential equation and its initial or boundary conditions.

## 3. Why is it important to calculate differential equations?

Differential equations are important because they allow us to model and understand complex systems and phenomena. They are used in many scientific and engineering fields to make predictions and solve real-world problems.

## 4. What are the applications of differential equations?

Differential equations have numerous applications in physics, chemistry, biology, economics, and engineering. They are used to describe the motion of objects, growth and decay of populations, heat transfer, chemical reactions, and many other phenomena.

## 5. Can differential equations be solved analytically?

Yes, some differential equations can be solved analytically, meaning that an explicit formula can be found for the solution. However, there are also many differential equations that can only be solved numerically using computer algorithms.

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