Separation of Variables - Arbitrary Constant, 14

Category: Differential Equations

"Published in Newark, California, USA"

Find the particular solution for

when x = x₀; v = v₀.

Solution:

Consider the given equation above  

By separation of variables, transfer dx to the right side of the equation as follows 

Integrate on both sides of the equation, we have 

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

Therefore, the particular solution is 

Separation of Variables - Arbitrary Constant, 13

Category: Differential Equations

"Published in Newark, California, USA"

Find the particular solution for

when θ = 0, r = a.

Solution:

Consider the given equation above  

By separation of variables, transfer r³ to the left side of the equation as follows  

Integrate on both sides of the equation, we have  

Substitute the values of r and θ 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