Separation of Variables - Arbitrary Constant, 7

Category: Differential Equations

"Published in Vacaville, California, USA"

Find the particular solution for

when x = 2; y = 3.

Solution:

Consider the given equation above   

Express y' as dy/dx as follows

By separation of variables, transfer x and dx to the right side of the equation and 1 + y² to the left side of the equation as follows 

Integrate on both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation, we have 

Substitute the values of x and y in order to solve for C as follows 

Therefore, the particular solution is 

Separation of Variables - Arbitrary Constant, 6

Category: Differential Equations

"Published in Newark, California, USA"

Find the particular solution for

when t = 0; r = r₀.

Solution:

Consider the given equation above  

By separation of variables, transfer dt to the right side of the equation and r to the left side of the equation as follows

Integrate on both sides of the equation, we have  

Substitute the values of r and t in order to solve for C as follows

Therefore, the particular solution is

Take the inverse natural logarithm on both sides of the equation, we have

Separation of Variables, 47

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above 

  
Divide both sides of the equation by y ln y as follows

Integrate on both sides of the equation, we have  

Therefore, the general solution is