How Do You Solve These Differential Equations Using Separation of Variables?

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SUMMARY

The discussion focuses on solving differential equations using the separation of variables technique. The two specific equations addressed are dy/dx = 1/y and xdy/dx = y. The procedure involves rearranging the equations to isolate variables, followed by integrating both sides. For the first equation, the transformation leads to ∫dy = ∫y dx, while the second requires multiplying by dx to yield dy = y dx.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically integration.
  • Familiarity with differential equations and their terminology.
  • Knowledge of algebraic manipulation techniques.
  • Ability to interpret dy/dx as a derivative of y with respect to x.
NEXT STEPS
  • Practice solving additional differential equations using separation of variables.
  • Learn about integrating factors for solving first-order differential equations.
  • Explore the method of exact equations in differential equations.
  • Study applications of differential equations in real-world scenarios.
USEFUL FOR

Students learning differential equations, educators teaching calculus, and anyone seeking to enhance their problem-solving skills in mathematics.

fran1942
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Hello, I have just started learning differential equations.
I am stuck on solving these two by separating the variables and writing the general solution.
Can someone please show me the procedure.

1. dy/dx = 1/y

2. xdy/dx = y

Thanks kindly for any help.
 
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Think the dy and dx as functions of x and y respectively, so dy/dx is just an ordinary division. Separation of variables is just the same as algebra - you need to get all the terms with a y on one side of the equals sign and all the terms with an x on the other side.

so for:
dy/dx=x you multiply both sides by dx to get dy=x.dx

then you just write an integration sign in front of each expression.
∫dy = ∫x.dx

now you follow that procedure for your examples.
 

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