Stuck in solving a (rather simple) differential equation problem.

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SUMMARY

The discussion centers around solving the differential equation dy/dx = x + (1/3)y^2, where y > 0. The participant is tasked with demonstrating that y = (1/2x^2 + x + 8)^(1/3) is a solution. After initial attempts to manipulate the equation, the user receives guidance to differentiate the proposed solution and substitute it back into the original equation to verify its validity. This approach emphasizes the importance of recognizing potential solutions in solving differential equations.

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  • Understanding of differential equations and their solutions
  • Knowledge of differentiation techniques
  • Familiarity with algebraic manipulation of equations
  • Basic calculus concepts, particularly related to functions and their derivatives
NEXT STEPS
  • Practice differentiating polynomial functions and applying the chain rule
  • Study methods for verifying solutions to differential equations
  • Explore the concept of implicit differentiation in relation to differential equations
  • Learn about specific types of differential equations and their solution techniques
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Students studying calculus, particularly those focusing on differential equations, as well as educators seeking to enhance their teaching methods in this area.

jowjowman
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Homework Statement


Equation: dy/dx=x+1/3y^2 where y>0


Homework Equations


I'm to show that (1/2x^2+x+8)^1/3 is a solution


The Attempt at a Solution


Splitting it up in x/3y^2+1/3y^2 has been futile and I'm out of ideas.

Can anyone help me see the light?
 
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jowjowman said:

Homework Statement


Equation: dy/dx=x+1/3y^2 where y>0
Is this 1+ (1/3)y^2 or 1+ 1/(3y^2) or (x+ 1)/(3y^2)?


Homework Equations


I'm to show that (1/2x^2+x+8)^1/3 is a solution
So you are NOT required to solve the equation? If y= ((1/2)x^2+ x+ 8)^(1/3), what is y'? Of course, y^2= ((1/2)x^3+ x+ 8)^(2/3). Put those into the equation and show that the the equation is satisfied.


The Attempt at a Solution


Splitting it up in x/3y^2+1/3y^2 has been futile and I'm out of ideas.

Can anyone help me see the light?
 
Yes, like many other problems in math, it is easier to solve a differential equation when you know the solution.

That may seem like cheating, but actually a rather well kept secret is that the only way to solve a differential equation is to know the solution, at least in outline.

Perhaps this concept can be further generalised. :shy:
 
Thanks HallsofIvy, you saved my day.
 

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