Free counters!

Thursday, March 27, 2014

Solving 2nd Order Differential Equations, 5

Category: Differential Equations, Integral Calculus, Algebra

"Published in Newark, California, USA"

Find the general solution for


Solution:

Consider the given equation above


The given equation is a 2nd Order Differential Equation because the second derivative of y with respect to x is involved. We can rewrite given equation as follows 



Multiply both sides of the equation by dx, we have




Integrate on both sides of the equation, we have 



Since the degree of a variable for both numerator and denominator are the same, then we have to do the division of a polynomial with another polynomial. After the division, the right side of the equation becomes








Multiply both sides of the equation by dx, we have 




Integrate on both sides of the equation, we have 




Consider


If


then


If


then


Hence, by integration by parts



Since the degree of a variable for both numerator and denominator are the same, then we have to do the division of a polynomial with another polynomial. After the division, the right side of the equation becomes

 

  



Substitute the above equation to the original equation, we have






 

Wednesday, March 26, 2014

Solving nth Order Differential Equations

Category: Differential Equations, Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Find the general solution for


Solution:

Consider the given equation above


The given equation is a 4th Order Differential Equation because the fourth derivative of y with respect to x is involved. We can rewrite the given equation as follows



Multiply both sides of the equation by dx, we have




Integrate on both sides of the equation, we have 




Rewrite the above equation as follows 



Multiply both sides of the equation by dx, we have 




Integrate on both sides of the equation, we have 






Rewrite the above equation as follows 



Multiply both sides of the equation by dx, we have 








Multiply both sides of the equation by dx, we have 








where