Differential Equations    02/20/16

During one’s study of calculus, one may encounter a peculiar set of derivatives known as Differential equations. Differential equations ae equations in which the derivative of the function is proportional to the function itself, examples including dydx=kyand dydx=y+x. These equations can not be solved through ordinary integration since we would be integrating in respect to a separate variable (yand dx), so one must use a method called separating and integrating, which involves moving the dxto the right side and all of the y’s to the left side (our first example would become dyy=kdx)and then integrate in respect to their variables. This is how the Exponential function comes into existence. Differential equations have an extensive range of application to other branches of science, particularly Physics, Economics, Biology, and Engineering. In fact, Differential Equations are so useful that there is an entire class at Universities dedicated to them that all Physics and Engineering majors must take!

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

Bernoulli Differential Equation   02/17/16

Bernoulli Differential Equations are a particular subset of FODes that have the form Y’+P(x)Y=Q(x)*Yn. By substituting (y)=1-Y1-N as the integrating factor and multiplying it out, we will be able to solve for the differential equation

Power Homogeneous DE    02/16/16
Power homogeneous Differential equations are differential equations that can be written as a function of yx. One can easily recognize a power homogeneous Differential equation if it is written in the form y’= G(x,y)H(x,y). If the equation is in this form, then we can make a substitution v= yxand then put in all in terms of Y.

First order Linear Differential Equations   02/15/16
First order Linear Differential equations are some of the most fundamental types in equations found in an advanced calculus course. Differential Equations can be summarized in the form dydt +p(t)y=g(t). F.O.D.Es can be solved by using a handy teqnique known as the integrating factor, or (t) = ep(t). By multiplying both sides of the equation by this constant, the left hand side will be in the form of a result of a product rule derivative, and then can be neatly integrated