Not sure what to do with this DE problem

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SUMMARY

The discussion focuses on verifying the solution of the differential equation y' = 3x² with the function y = x³ + 7. The verification process involves substituting y into the derivative equation, resulting in (x³ + 7)' = 3x². The left-hand side simplifies to 3x², confirming that the function is indeed a solution. The participant expresses confusion over the necessity of this verification step, which is typically straightforward for students.

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Homework Statement


Verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x.
y'=3x2; y=x3+7


Homework Equations





The Attempt at a Solution


Well obviously I see that the derivative of y=x3+7 is just 3x2, but what does it mean by verifying by substitution? :confused:
 
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iRaid said:

Homework Statement


Verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to x.
y'=3x2; y=x3+7

Homework Equations


The Attempt at a Solution


Well obviously I see that the derivative of y=x3+7 is just 3x2, but what does it mean by verifying by substitution? :confused:

It means substitute y = x^3 + 7 into y' = 3x^2 to get (x^3 + 7)' = 3x^2 and then confirm that the left hand side does in fact equal the right hand side. In this case it obviously does, so there's nothing more to do. Although I suppose you could expressly state that (x^3 + 7)' = (x^3)' + (7)' = 3x^2 + 0 = 3x^2.
 
Why do they make me even do this...?
 
Usually, they don't anticipate the student having any problem verifying an equation given its solution.
 

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