Day: February 19, 2016

Exact Differential Equations

Exact Differential Equations

Exact Differential Equations 02/19/16

 

During a course of study through Differential Equations, one may encounter a set of problems known as exact differential equations. Exact Differential equations are usually from a result of a differentiation of a multivariable function, and usually come in the form M(x,y)+N(x,y)dydx=f(a,b)’=0 with f(a,b)=c and M(x,y)is assumed by be the partial with respect to x and N(x,y)is assumed to be the partial in respect to y. One can check if this is an Exact DE by differentiating M in respect to y and N in respect to x and checking if it’s valid (Since f(x,y)xy=f(x,y)yx). If these are equal, we then integrate M(x,y)in respect to x to get back f(x,y). However, instead of a constant, we use a function of y h(y)since differentiating a y-based function in respect to x will nullify it. We then derive our new f(a,b)in respect to y and set f(a,b)’=n(x,y)and use the remainder to find h(x,y)’. Once we do that, we integrate h(a,b)’with respect to y and then add it to our equation and set it all equal to 0 to complete everything. That is how we solve such equations