Differential equation substituition of new terms

Thread starter delsoo
Start date May 4, 2014
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Differential Differential equation Terms

In summary, differential equation substitution of new terms is a process that involves introducing new variables in a differential equation to simplify its form and make it easier to solve. This can be necessary as it reduces the complexity of the equation and allows for the use of methods not applicable to the original form. Common substitution techniques include trigonometric, power, and logarithmic substitutions. Substituting new terms can also change the solution of a differential equation, as the new terms can alter the form and methods used to solve it. This technique can be applied in various fields to model real-world problems and make predictions or analyze systems.

May 4, 2014

delsoo

Homework Statement

for this question, i sub u=xy and try to eliminate y in my working.. but how should i proceed since the term ux cannot be separated

Homework Equations

The Attempt at a Solution

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May 4, 2014

ehild

Homework Helper

I do not understand what you did. Your u-s and y-s look the same. Eliminate y by replacing it with u/x. Note that xdy/dx+y=du/dx, and xy²=u²/x.

ehild

1. What is differential equation substitution of new terms?

Differential equation substitution of new terms is a process in which new variables are introduced in a differential equation to simplify its form and make it easier to solve. This allows for more complex equations to be solved using methods that are not applicable to the original form.

2. Why is it necessary to substitute new terms in a differential equation?

Substituting new terms in a differential equation can make it easier to solve by reducing the complexity of the equation. This can also help in finding a general solution that can be used for a wider range of problems.

3. What are some common substitution techniques used in differential equations?

Some common substitution techniques used in differential equations include trigonometric substitutions, power substitutions, and logarithmic substitutions. These techniques can be used to simplify the equation and make it easier to solve.

4. Can substitution of new terms change the solution of a differential equation?

Yes, substitution of new terms can change the solution of a differential equation. This is because the new terms introduced can change the form of the equation and the methods used to solve it, resulting in a different solution.

5. How can differential equation substitution of new terms be applied in real-world problems?

Differential equation substitution of new terms can be applied in various fields such as physics, engineering, and economics to model real-world problems. By introducing new terms, the equations can be simplified and solved to find solutions that can be used to make predictions or analyze systems in the real world.