Verifying a Solution for a Basic Differential Equation

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SUMMARY

The discussion focuses on verifying that the function y = x - (x^-1) is a solution to the differential equation xy' + y = 2x. Participants emphasize the importance of substituting the proposed solution into the equation to confirm its validity rather than solving the differential equation itself. Key techniques discussed include substitution of y and its derivative y' into the equation to check for equality. This approach simplifies the verification process and clarifies the solution's correctness.

PREREQUISITES
  • Understanding of basic differential equations
  • Familiarity with function substitution techniques
  • Knowledge of derivatives and their notation
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study the method of substitution in differential equations
  • Learn about verifying solutions to differential equations
  • Explore techniques for separating variables in differential equations
  • Review the properties of linear differential equations
USEFUL FOR

Students studying differential equations, educators teaching calculus concepts, and anyone interested in mathematical problem-solving techniques.

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Homework Statement
Show that y = x-(x^-1) is a solution for the differential equations:
xy' +y = 2x

The attempt at a solution
Frankly, I haven't a clue how to start.

If it weren't for the x in the xy' term this would be easy, since I could just integrate; unfortunately, I don't know how to separate the x's from the y's, and this is really a problem, because this is the first problem from the homework set. I re-read the textbook, but don't get it. Any suggestions for techniques to separate the variables?
 
Physics news on Phys.org
The problem isn't asking you to solve the differential equation, merely to verify that the solution given is in fact a solution. So, substitute it in for y and y' and see if the equation holds true.
 

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