In the previous article, you learned how to solve simple differential equations. These equations can be linear or non-linear, and can have a range of solutions.
While some of these solutions are interesting and valuable in teaching students about different areas of science, technology, and math, others are not so helpful.
For example, a linear equation that does not have a re-solution for x2 + 2x = 0 does not mean it is unhelpful. Many times, it is the fact that the equation has a negative value for x that causes it to be unhelpful.
Put it into the form Q(x) + R(x)
When there is a equation with more than one variable, you can use the fact that there are two unknowns to suggest a general solution.
The equation has two variables, so you can use the familiar formula for finding the square of a number: X2 + Y2 = 32.
By using this formula, you can find the general solution to the given differential equation. The trick is to find the source of the 2nd and 3rd terms in the equation.
By doing this, you create a new term in your solution, making it more stable. You may have to look at several equations to find the source of the 2nd and 3rd terms!
This tip is about solving differential equations that have more than one variable.
Find the general solution
When solving a differential equation, you need to find the asymptotes and general solution. As the asymptotes are points where the equation becomes zero, these can be found easily.
The general solution requires finding a change in variables that will create a new problem and solve it. This can be very difficult, so most savvy algebra teachers would make students do this first before trying to find the asymptotes.
It takes more work to find the general solution, but it is worth it in the end.
Check for extraneous solutions
If your differential equation has a very complicated general solution, check for extraneous solutions.
Many times, these solutions are wrong, and will not be the solution to the differential equation. These solutions may be circles or other shapes that fit in the space given to create an artificial balance to the space.
These extraneous solutions may also be inappropriate, such as solving the equation in circles or on a circle. As previously mentioned, this is a common cause of singularity problems in equations.
General solutions do not have to be excluded always! It is important to look for them when they are provided as clues.
Check for universality
If your differential equation does not have a general solution, try to find a similar differential equation with the same variable.
If you do not see a similar equation with the same variable, your problem is not as difficult as it may seem. There are many software programs that can find similar equations and solutions to differential equations.
However, before you use this information to solve your problem, make sure that the software you use has been tested for this purpose. Many times, software does not have the expertise to find solutions that are more universal than your original problem.
Solve for x and y in your software to find the general solution of the given differential equation. If those values are not valid, go ahead and change one or both of those variables to find the general solution.
Use software to find the solution
Some differential equations have very complicated general solutions.
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