The differential equation x22yx2−7xyx+16y=0
has x4 as a solution.
Applying reduction order we set y2=x4.
Then (using the prime notation for the derivatives)
y′2=
y′′2=
So, plugging y2 into the left side of the differential equation, and reducing, we get
x2y′′2−7xy′2+16y2=
The reduced form has a common factor of x5 which we can divide out of the equation so that we have x′′+′=0.
Since this equation does not have any u terms in it we can make the substitution =′ giving us the first order linear equation x′+=0.
This equation has integrating factor
for x > 0.
If we use a as the constant of integration, the solution to this equation is =
Integrating to get u, and using b as our second constant of integration we have =
Finally y2=
and the general solution is

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For this option, you will work individually. the pythagorean theorem can be used in many real-world scenarios. part 1 write your own real-world scenario where the pythagorean theorem can be applied to find a missing piece. you may choose to write a problem that is two- or three-dimensional in nature. be sure that you will be able to draw a diagram of your scenario. write out your problem and submit it for part 1. be sure to end your scenario with a question. part 2 draw a diagram of the scenario you created in part 1. you may draw by hand and scan and upload your drawing or create a computer-generated drawing for submission. be sure to label all parts and dimensions of the drawing. part 3 solve the question that you posed in part 1. show all of your steps in answering the question. for this option, you will need to submit all three parts for full credit—your real-world problem and question, the diagram that you created, and your work solving the problem, showing all steps. * note that your instructor is looking for your own original idea. while it is acceptable to use the internet for research and inspiration, academic integrity policies apply.