Category: Differential Equations
"Published in Newark, California, USA"
Find the general solution for
Solution:
Consider the given equation above
Did
you notice that the given equation cannot be solved by separation of
variables? The trigonometric function is a combination of x and y in the group and
there's no way that we can separate x and y.
This
type of differential equation is a homogeneous function. Let's consider
this procedure in solving the given equation as follows
Let
so that
Substitute the values of y and dy to the given equation, we have
The resulting equation can now be separated by separation of variables as follows
Integrate on both sides of the 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
Category: Differential Equations
"Published in Newark, California, USA"
Find the general solution for
Solution:
Consider the given equation above
Transfer all the terms from the right side of the equation to the left side of the equation and group as follows
In order to separate dx and dy from other variables, divide both sides of the equation by (1 - y²)(1 + x²) as follows
Integrate both sides of the equation, we have
Next, we need to solve for the value of A and B by partial fractions as follows
Equate the coefficients for y, we have
Equate the coefficients for y0, we have
Substitute the value of A and B to the above equation, we have
Take the inverse natural logarithm on both sides of the equation, we have
where D = C².
Therefore, the general solution is
Category: Differential Equations
"Published in Newark, California, USA"
Find the general solution for
Solution:
Consider the given equation above
In order to separate dx and dy from other variables, divide both sides of the equation by cos² x cos² y as follows
Integrate both sides of the equation, we have
Therefore, the general solution is
