Differential Equations by separation of variables

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SUMMARY

The discussion focuses on solving the differential equation dy/dx = x*(1 - y^2)^(1/2) using the method of separation of variables. Participants emphasize the importance of rearranging the equation to isolate dy and dx, leading to the expression (1/sqrt(1 - y^2)) * dy = x dx. The integration of both sides is then suggested as the next step to find the solution in the form of y. This method is a standard approach in solving first-order differential equations.

PREREQUISITES
  • Understanding of first-order differential equations
  • Familiarity with the separation of variables technique
  • Basic integration skills
  • Knowledge of trigonometric identities related to y = sin(theta)
NEXT STEPS
  • Practice solving additional differential equations using separation of variables
  • Explore integration techniques for more complex functions
  • Study the implications of initial conditions on differential equation solutions
  • Learn about the graphical interpretation of solutions to differential equations
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in mastering differential equations and their applications.

LAK
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Can someone please help me to calculate the following using separation of variables:

dy/dx = x*(1 - y^2)^(1/2)

to that the solution is in the form:

y =
 
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I have moved both your threads here to the Differential Equations subforum as this is a better fit for them.

What do you get when you separate the variables, before integrating?
 
LAK said:
Can someone please help me to calculate the following using separation of variables:

dy/dx = x*(1 - y^2)^(1/2)

to that the solution is in the form:

y =

For starters:

$\displaystyle \begin{align*} \frac{1}{\sqrt{1 - y^2}} \, \frac{dy}{dx} = x \end{align*}$

and now you can integrate both side w.r.t. x :)
 

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