Solution of exact differential equation

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SUMMARY

The discussion centers on solving an exact differential equation, specifically addressing a student's confusion regarding the disappearance of part of the equation during the solution process. The key insight provided is the application of the product rule in differentiation, particularly when y is treated as a function of x. The relevant equation presented is the derivative of the product of an exponential function and y, expressed as \(\frac{d}{dx}(e^{3x}y) = 3e^{3x}y + e^{3x}\frac{dy}{dx}\). This clarification enables the student to progress in solving the differential equation.

PREREQUISITES
  • Understanding of exact differential equations
  • Knowledge of the product rule in calculus
  • Familiarity with differentiation of exponential functions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the concept of exact differential equations in detail
  • Review the product rule and its applications in calculus
  • Practice solving various types of differential equations
  • Explore the implications of treating variables as functions of one another
USEFUL FOR

Students preparing for exams in calculus, particularly those focusing on differential equations, as well as educators seeking to clarify concepts related to exact derivatives and differentiation techniques.

adam640
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Hi, I'm looking at a past paper for an exam I have on Tuesday and I'm struggling to understand this question on exact derivatives.

Here is a link to the question: http://imageshack.us/photo/my-images/812/questiong.png/

I have looked over my notes for guidance and eventually turned to the solution in the hope that I could work backwards. However I do not understand why half of the equation seems to have disappeared?

Here is the first stage of the solution: http://imageshack.us/photo/my-images/221/solutionp1cvsg.png/

If anyone could help explain to me why this is the case it would be greatly appreciated. I am confident that I would be able to solve the question from here.

Thanks,

Adam
 
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It's just the product rule, where y is a function of x.

<br /> \frac{d}{dx}(e^{3x}y) = 3e^{3x}y + e^{3x}\frac{dy}{dx}<br />
 

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