Solving Separable Differential Equation with Initial Condition

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SUMMARY

The discussion centers on solving the separable differential equation dy/dx = sqrt(4y+64) with the initial condition y(4)=9. The user attempts to manipulate the equation into a form suitable for integration but expresses uncertainty about the integration process. The correct approach involves separating variables and integrating both sides, leading to the particular solution that satisfies the given initial condition.

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  • Understanding of separable differential equations
  • Knowledge of integration techniques
  • Familiarity with initial value problems
  • Basic algebraic manipulation skills
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  • Study the method of separation of variables in differential equations
  • Learn integration techniques for functions involving square roots
  • Explore initial value problems and their solutions
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Students and professionals in mathematics, particularly those studying differential equations, as well as educators looking for examples of solving initial value problems.

alchal
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QUESTION:

Solve the separable differential equation
dy/dx = sqrt(4y+64), Initial Condition: y(4)=9,
and find the particular solution satisfying the initial condition.

MY ATTEMPT:

(dy/dx)^2 = 4y+64
((dy/dx)^2)-4y = 64
,/' (((dy/dx)^2)-4y) dx = ,/' 64 dx

Is this the right method? If so, not sure how to integrate (dy/dx)^2

Any suggestions?
 
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dy/sqrt(4y+64) = dx
 
Thank you!
 

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