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

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In summary, the conversation is about a person seeking help with solving two differential equations by separating the variables and writing the general solution. They are given guidance on the procedure, which involves getting all terms with y on one side and all terms with x on the other, then integrating both sides. This process is demonstrated for the given equations.
  • #1
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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  • #2
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.
 

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

1. How do I solve two simple equations?

To solve two simple equations, you need to follow a few steps. First, identify the variables in each equation. Then, isolate one variable by using inverse operations. Finally, substitute the value of the isolated variable into the other equation to solve for the other variable.

2. Can you provide an example of solving two simple equations?

Sure, let's say we have the equations 3x + 2 = 8 and 2x - y = 6. First, we identify the variables x and y. Then, we isolate x in the first equation by subtracting 2 from both sides, giving us 3x = 6. Next, we divide both sides by 3 to get x = 2. Finally, we substitute x = 2 into the second equation to solve for y, giving us y = -2.

3. What if there are fractions or decimals in the equations?

If there are fractions or decimals in the equations, you can eliminate them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This will result in whole numbers in the equation, making it easier to solve.

4. What if there are variables on both sides of the equations?

If there are variables on both sides of the equations, you can still solve them by isolating one variable on one side of the equation and then using substitution to solve for the other variable. It may involve more steps, but the process remains the same.

5. Are there any shortcuts or tricks to solving two simple equations?

Yes, there are some common methods like elimination or substitution that can be used to solve two simple equations. However, it is important to understand the concept of equations and how to use inverse operations to get the correct solution.

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