Exact Equation - Arbitrary Constant

Category: Differential Equations, Integral Calculus, Differential Calculus

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Solve for the particular solution for

when x = 0, y = 2.

Solution:

The first that we have to do is to check the above equation if it is exact or not as follows

Let

then

Let

then

Since

then the given equation is Exact Equation. The solution for the above solution is F = C. Consider the given equation

Let

and

Integrate the partial derivative of the first equation above with respect to x, we have

Since we are integrating the partial derivatives, then another unknown function, T(y) must be added. If

then

To solve for T'(y), equate

To solve for T(y), integrate on both sides of the equation with respect to y, as follows

Since the arbitrary constant is already included in F = C, then we don't have to add the arbitrary constant in the above equation. Therefore,

The general solution of the equation is

To solve for the value of C, substitute x = 0 and y = 2 to the above equation, we have

Therefore, the particular solution of the equation is