# Ordinary Differential equations question

• Hutchyy
In summary, the conversation discusses how for a third order constant coefficient homogeneous differential equation with solutions y1, y2, and y3, if the Wronskian of y1 and y2 is always positive, but the Wronskian of all three solutions is 0 when t=0, then there exists constants c1 and c2 such that c1y1(t) + c2y2(t) = y3(t) for all real t. The significance of plugging in 0 for t in the Wronskian is related to Abel's theorem, but the connection between a positive Wronskian and a Wronskian with 0 when t=0 is not clear. The conversation also
Hutchyy
#17 If you can't see the picture: Suppose that y1, y2, and y3 are solutions to a third order constant coefficient homogeneous differential equation. Suppose further that for all real t, W(y1,y2)(t)>0, but also W(y1,y2,y3)(0)=0. Then there exists c1 and c2 such that c1y1(t) + c2y2(t) =y3(t) for all real t. Is this true, false or maybe true? I know it must have something to do with abel’s theorem but I can’t really figure out how it applies.. ,

what is the significance of plugging in 0 for ’t’ in the wronskian?

I'm thinking its something that has to do with abel's theorem but I can't make any connections as to how having a positive wronskian relates to a bigger wronskian with 0 plugged into equalling 0. https://www.physicsforums.com/attachments/90663

There's something wrong with your attachment. I can't view it.

When y= 0 (along the x-axis), dy/dx= 0. Which of the direction fields has that property? When x= 0 (along the y axis) dy/dx= y^2. Which of the direction fields has that property?

Thanks halls of ivy! But I got that one! I meant number 17 the word problem

## 1. What is an ordinary differential equation (ODE)?

An ordinary differential equation is a mathematical equation that describes how a function changes over time. It involves a function and its derivatives, and is used to model various physical phenomena in fields such as physics, engineering, and biology.

## 2. What are the different types of ODEs?

There are several types of ordinary differential equations, including:
- Separable ODEs
- Exact ODEs
- Linear ODEs
- Nonlinear ODEs
- First-order ODEs
- Second-order ODEs
- Higher-order ODEs
Each type has its own unique characteristics and methods for solving.

## 3. What is the difference between an initial value problem and a boundary value problem for ODEs?

An initial value problem involves finding a solution to an ODE at a specific initial value, while a boundary value problem involves finding a solution that satisfies certain conditions at different points along the domain of the function. In other words, an initial value problem has one known value, while a boundary value problem has multiple known values.

## 4. How are ODEs used in real-world applications?

ODEs are used in a variety of real-world applications, such as modeling population growth, predicting the motion of objects, and analyzing chemical reactions. They are also used in engineering to design systems and control processes, and in economics to model supply and demand.

## 5. What methods are used to solve ODEs?

There are several methods for solving ODEs, including:
- Analytical methods, such as separation of variables and integrating factors
- Numerical methods, such as Euler's method and Runge-Kutta methods
- Series solutions, such as power series and Fourier series
The choice of method depends on the type of ODE and the desired level of accuracy.

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