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Friday, March 27, 2015

Separation of Variables, 24

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 cos x sin y, we have




Integrate on both sides of the equation, we have 






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





Therefore, the general solution is 

Thursday, March 26, 2015

Separation of Variables, 23

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for


Solution:

Consider the given equation above


Express y' as dy/dx as follows



Multiply both sides of the equation by dx, we have




Since both sides of the equations are already factored according to their variables, then we can separate the other variables from dy and dx as follows



Integrate on both sides of the equation, we have







             
                 

Therefore, the general solution is


where D = 1/C.

Wednesday, March 25, 2015

Eliminating Arbitrary Constant, 25

Category: Differential Equations

"Published in Newark, California, USA"

Eliminate the arbitrary constant for


Solution:

Consider the given equation above  


Take the derivative on both sides of the equation with respect to x, we have    





Take the derivative on both sides of the equation with respect to x, we have    





Multiply the first derivative equation by -2 and add it to the second derivative equation, we have


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Substitute the value of c1 to the second derivative equation, we have 












Substitute the values of c1 and c2 to the given equation and therefore, the final answer is