Separation of Variables, 42

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above  

Expand the given equation and group according to their variables as follows

Divide both sides of the equation by αβ as follows 

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

Separation of Variables, 41

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 dx and (e^2x + 4) to the right side of the equation and y to the left side of the equation, we have  

Integrate on both sides of the equation, we have  

Did you notice that the right side of an equation cannot be integrated by simple integration? The differential of e^2x + 4 is 2e^2xdx and 2e^2x is missing at the numerator. Because of this, we have to use the algebraic substitution as follows

Let  

So that 

The value of dx is  


Hence, the above equation becomes

By separating the partial fractions, we can solve for the values of A and B as follows

Equate x⁰:

                           

Equate x:

                    

Substitute the values of A and B to the above equation, we have

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

but 


Hence, the above equation becomes

Therefore, the general solution is