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Find the general solution for the equation:
Solution:
If you examine the above equation, it is a differential equation because it contains the differentials like dx and dy. Our goal is to eliminate the differentials by integration but before that, let's examine the above equation. We have to do first the grouping at the right side of the equation and then factor all the terms if there are any common factors. The above equation can be written as
You noticed that we are applying the principles of factoring for the above equation. You must know the principles of Algebra, Trigonometry, and Calculus very well in solving Differential Equations. Next, we want to eliminate y at the left side of the equation and also we want to eliminate x at the right side of the equation. If you will divide both sides of the equation by (y + 1)(x2 +1), then the above equation becomes
Now, looking at the right side of the equation, the exponential degree at the numerator is greater than at the denominator. We have to do the division first as follows
The above equation can be written as
Next, integrate on both sides of the equation
Multiply both sides by 2 to eliminate the fraction, we have
A constant (C) multiply by any number is still a constant.
Note: In some cases, C can be written as ln C after the integration. It's up to you. The natural logarithm of any constant is still a constant.