General solution to a second order homogeneous differential equation

In summary, a second order homogeneous differential equation is a mathematical equation that relates the second derivative of an unknown function to the function itself and does not contain any constant terms. Its general solution includes all possible solutions to the equation, with two arbitrary constants that can be adjusted to fit specific initial conditions. To find the general solution, the auxiliary equation must be solved and its roots used to form the general solution. A second order homogeneous differential equation can have an infinite number of general solutions, but each specific initial condition will have a unique solution. A particular solution is a specific solution that satisfies given initial conditions, while the general solution includes all possible solutions and can be obtained by substituting specific values for the arbitrary constants.
  • #1
KaiserBrandon
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0

Homework Statement



Find if it is true that the general solution to : y'' - y' = 0, where y(x),
can be written as : y(x) = c1 cosh(x) + c2 sinh(x), where c1 and c2 are real
arbitrary constants.

Homework Equations



differential equation solving

The Attempt at a Solution



I just want to know what the h's mean at the end of sin and cos.
 
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  • #2
The 'h' means hyperbolic. cosh(x)=(e^x+e^(-x))/2. sinh(x)=(e^x-e^(-x))/2. That's all, it's just an abbreviation for those expressions. Can you show they both satisfy your differential equation?
 

1. What is a second order homogeneous differential equation?

A second order homogeneous differential equation is a mathematical equation that relates the second derivative of an unknown function to the function itself. It is called "homogeneous" because the equation is equal to zero, meaning that it contains no constant terms.

2. What is a general solution to a second order homogeneous differential equation?

A general solution to a second order homogeneous differential equation is an expression that includes all possible solutions to the equation. It contains two arbitrary constants that can be adjusted to fit specific initial conditions.

3. How do you find the general solution to a second order homogeneous differential equation?

To find the general solution, you first need to solve the auxiliary equation, which is obtained by setting the coefficient of the second derivative term to zero. Then, you can use the roots of the auxiliary equation to form the general solution.

4. Can a second order homogeneous differential equation have more than one general solution?

Yes, a second order homogeneous differential equation can have an infinite number of general solutions. This is because it contains two arbitrary constants that can take on any value. However, each specific initial condition will only have one unique solution.

5. What is the difference between a particular solution and a general solution to a second order homogeneous differential equation?

A particular solution is a specific solution to the differential equation that satisfies given initial conditions, while a general solution includes all possible solutions to the equation. A particular solution can be obtained by substituting specific values for the arbitrary constants in the general solution.

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