# System of Equations for Second-Order IVP

• MHB
• karush
In summary, to change the second-order IVP $y''+y'-2y=0, y(0)=2, y'(0)=0$ into a system of equations, we can let $x_1=y$ and $x_2=y'$ and substitute to get the system of first order equations $x_1'=x_2, x_2'=-x_2+2x_1$. From this, we can determine that $x_1(0)=y(0)=2$ and $x_2(0)=y'(0)=0$. However, it can be confusing to see multiple examples with different substitutions.
karush
Gold Member
MHB
Change the second-order IVP into a system of equations
$y''+y'-2y=0\quad y(0)= 2\quad y'(0)=0$
let $x_1=y$ and $x_2=y'$ then $x_1'= x_2$ and $y''=x_2'$
then by substitution
$x_2'+x_2-2x_1=0$
then the system of first order of equations
$x_1'=x_2$
$x_2'=-x_2+2x_1$

hopefully so far..

Yes, that is correct.

Now, what are $x_1(0)$ and $x_2(0)$?

Country Boy said:
Yes, that is correct.
Now, what are $x_1(0)$ and $x_2(0)$?
$x_1'=x_2=y(0)=0$
and
$x_2'=-x_2+2x_1=0+2(2)=4$

its like chasing a rabbit in the briers

Frankly, I am not sure what you are doing, You were told that y(0)= 2 and y'(0)= 0.

Since you defined $x_1(t)$ to be y(t) and $x_2(t)$ to be y'(t),
$x_1(0)= y(0)= 2$ and $x_2(0)= y'(0)= 0$

Country Boy said:
Frankly, I am not sure what you are doing, You were told that y(0)= 2 and y'(0)= 0.

Since you defined $x_1(t)$ to be y(t) and $x_2(t)$ to be y'(t),
$x_1(0)= y(0)= 2$ and $x_2(0)= y'(0)= 0$

i think I get confused looking at multiple examples with all these different substitutions

## What is a system of equations for second-order initial value problems (IVPs)?

A system of equations for second-order IVPs is a set of two differential equations that involve a second derivative of a function and its initial conditions. These equations are used to model physical systems and can be solved to find the function that describes the behavior of the system over time.

## What is the difference between a first-order and second-order IVP?

The main difference is that a first-order IVP involves only one differential equation, while a second-order IVP involves two differential equations. This means that a second-order IVP has more information and can model more complex systems.

## How do you solve a system of equations for second-order IVPs?

To solve a system of equations for second-order IVPs, you can use various methods such as substitution, elimination, or matrices. These methods involve manipulating the equations to isolate the variables and find their values at different points in time.

## What are initial conditions in a system of equations for second-order IVPs?

Initial conditions are the values of the function and its derivatives at a specific point in time, usually denoted as t=0. These conditions are necessary to solve the system of equations and determine the behavior of the system over time.

## What are some real-world applications of systems of equations for second-order IVPs?

Systems of equations for second-order IVPs are used in various fields such as physics, engineering, and economics to model and predict the behavior of complex systems. They can be used to study the motion of objects, the growth of populations, and the flow of electricity, among others.

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