Separation of Variables, 37

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above 

Since the given equation is already arranged according to their variables, then we can integrate on both sides of the equation as follows

In this type of integration of trigonometric functions, we need to use the principles of trigonometric identities first before we can do the simple integration of trigonometric functions. Hence, the above equation becomes

Therefore, the general solution is 

where D = 3C. 

Separation of Variables, 36

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

By separation of variables, transpose x² dx to the right side of the equation and e^y to the left side of the equation, we have

Integrate on both sides of the equation, we have  

Therefore, the general solution is

where G = - C.

Separation of Variables, 35

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 y³e^-x in order to separate dx and dy from other variables as follows   

Integrate on both sides of the equation, we have 

Therefore, the general solution is