Solving a special type of a second order differential equation

In summary, the conversation discusses two types of second order differential equations and their corresponding methods for solving them. The problem at hand, however, uses the first type even though the variable x is present. The solution involves manipulating the equation to turn it into a first order equation in v.
  • #1
Jim4592
49
0

Homework Statement



y''[x] = y'[x] + x


Homework Equations


We were taught two special types of second order diff. equations:

Type 1: Supposed to be when x is missing
v = y'[x]
v'[x] = y''[x]

Type 2: Supposed to be when y is missing
v = y'[x]
v v'[x] = y''[x]


The Attempt at a Solution


The answer key reads:

v'[x] - v = x

d/dx{e^-x v} = x e^-x

e^-x v = C1 * e^-x -x -1

v = C1 * e^x -x -1

y[x] = c2 - c1 * e^-x - (.5)x^2 - x


I don't understand why we used type 1 to solve this problem since x is clearly stated in the problem. I was hoping someone could explain, thanks.
 
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  • #2
Are you sure you have listed the two types correctly? Usually, if y is missing you would do:

v = y'
v' = y''

so your second order equation in y becomes a first order equation in v. That is the type of equation you have and also the method your manual gives.
 

FAQ: Solving a special type of a second order differential equation

1. What is a second order differential equation?

A second order differential equation is a mathematical equation that involves the second derivative of a function. It is typically written in the form y'' = f(x,y,y'), where y is the dependent variable, x is the independent variable, and y' and y'' represent the first and second derivatives of y with respect to x, respectively.

2. What makes a second order differential equation special?

A second order differential equation is considered special because it contains the second derivative, which allows for more complex behavior and solutions compared to first order differential equations. It also has a wide range of applications in various fields such as physics, engineering, and economics.

3. How do you solve a second order differential equation?

There are several methods for solving a second order differential equation, including the method of undetermined coefficients, variation of parameters, and Laplace transforms. The specific method used depends on the form of the equation and the initial conditions given.

4. What is the importance of solving a second order differential equation?

Solving a second order differential equation allows us to understand the behavior and relationships of variables in a system. This is crucial in many scientific and engineering applications where accurate predictions and analysis are necessary.

5. Are there any real-world examples of second order differential equations?

Yes, there are many real-world examples of second order differential equations. Some examples include the motion of a mass on a spring, the motion of a pendulum, and the growth and decay of populations. These examples can be described by second order differential equations and have important applications in various fields.

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