Using reduction of order for Non-Homogenous DE

Thread starter juice34
Start date Apr 10, 2008
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In summary, reduction of order is a technique used to solve non-homogenous differential equations (DE). It involves reducing the order of the DE by substituting a new variable and solving for it, followed by finding the general solution of the original DE using the new variable. This method is particularly useful for non-homogenous DEs, where the right-hand side of the equation contains a non-zero function, making it difficult to solve using traditional methods. Reduction of order simplifies the equation and allows for a more straightforward solution.

Apr 10, 2008

juice34

I need help solving a higher order differential equation by reduction of order.
It will be greatly appreciated if all steps are posted as well!

y(Double Prime)-3y(Prime)+2y=5e^3x where y sub one =e^x.

Ive tried to get the answer but end up with 3 constants.

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Apr 10, 2008

juice34

the teachers example goes from w to u prime to u without integrating u prime to get u, where w=u prime and y=u*y(sub one)

FAQ: Using reduction of order for Non-Homogenous DE

What is a non-homogenous differential equation?

A non-homogenous differential equation is a type of differential equation in which the right-hand side of the equation contains a term that is not equal to zero. This term can be a function of the independent variable or a constant.

What is reduction of order for non-homogenous differential equations?

Reduction of order is a method used to solve non-homogenous differential equations by reducing the order of the equation. This involves introducing a new variable and substituting it into the original equation to create a new equation of lower order.

When should reduction of order be used for non-homogenous differential equations?

Reduction of order should be used when the non-homogenous differential equation cannot be solved using other methods such as separation of variables or substitution. It is also used when the non-homogenous term can be expressed as a product of a function and its derivative.

What are the steps for using reduction of order for non-homogenous differential equations?

The steps for using reduction of order are as follows:
1. Identify the non-homogenous term in the equation.
2. Introduce a new variable by letting u=yv, where y is the known solution to the corresponding homogeneous equation.
3. Substitute u into the original equation and solve for v.
4. Integrate v to obtain y.
5. Use the known solution y and the newly obtained solution v to find the general solution to the non-homogenous equation.

What are the limitations of using reduction of order for non-homogenous differential equations?

Reduction of order can only be used for non-homogenous differential equations with certain types of non-homogenous terms, such as a product of a function and its derivative. It also requires knowledge of the corresponding homogeneous solution, which may not always be easily obtainable. Additionally, this method may not always yield a general solution and may only provide a particular solution in some cases.