# Ordinary Differential Equations

• aaronfue
So the first equation can be rewritten as y = \pm\sqrt{C_1(x + \frac{C_1}{4})} In summary, the conversation discusses how to solve the equations y = 2xy' + y(y')^2 and y^2 = C1(x + C1/4). The individual is unsure of how to handle the C1 term in the second equation. It is suggested to take the derivative of y and plug it into the first equation. This results in the solution y = \pm\sqrt{C_1(x + \frac{C_1}{4})}.
aaronfue

## Homework Statement

y = 2xy' + y(y')2; y2 = C1 (x + C1 /4)

2. The attempt at a solution

I thought that I could take the first equation and set it equal to zero. But the C1 in the second equation is throwing me off.

Am I supposed to set the second equal to C1?

Last edited:
Are you saying that you want to show that $y= \pm\sqrt{C_1(x+ \frac{C_1}{4})}$ satisfies [/itex]y= 2xy'+ y(y')^2[/itex]?

Just find the derivative of y and plug it into the equation.

HallsofIvy said:
Are you saying that you want to show that $y= \pm\sqrt{C_1(x+ \frac{C_1}{4})}$ satisfies [/itex]y= 2xy'+ y(y')^2[/itex]?

Just find the derivative of y and plug it into the equation.

Sorry the second equation is y2, but I think I see what you are saying.

Yes, that's why I took the square root to get y.

## 1. What is an Ordinary Differential Equation (ODE)?

An Ordinary Differential Equation (ODE) is a mathematical equation that involves a function and one or more of its derivatives. It represents the relationship between a function and its rate of change over time or space.

## 2. What are some real-life applications of Ordinary Differential Equations?

ODEs are used to model a wide range of physical, biological, and social phenomena. Some examples include population growth, chemical reactions, pendulum motion, and electrical circuits.

## 3. How do you solve an Ordinary Differential Equation?

The method for solving an ODE depends on its type and complexity. Some common techniques include separation of variables, Euler's method, and Laplace transforms. In some cases, analytical solutions may not exist and numerical methods are used instead.

## 4. What is the difference between an Ordinary Differential Equation and a Partial Differential Equation?

The main difference between an ODE and a Partial Differential Equation (PDE) is the number of independent variables. ODEs involve only one independent variable, while PDEs involve two or more variables. PDEs are typically used to model phenomena involving multiple variables, such as heat transfer and wave propagation.

## 5. How are Ordinary Differential Equations used in machine learning and artificial intelligence?

ODEs are used in machine learning and artificial intelligence to model and predict systems that evolve over time. They can be used to simulate the behavior of complex systems and make predictions based on past data. ODEs are also used in optimization algorithms and neural networks to improve performance and accuracy.

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