First Order Differential Equation

A First Order Differential Equation is involving the unknown function y, its derivative y’ and the variable x. the most general type of  First Order Differential Equation can be written as Y’=f(x,y),or we can write as dy/dx=f(x,y), here f(x,y) is a function of two variables defined on a region in the xy plane. The equations is of first order because it involves only the first derivative y’ or dy/dx.

Now questions is that how do we solve first order differential equation. There are two methods which can be used to solve first order differential equation. First is separation of variables and second is Integrating factor. Again the new question in your mind should be how do we know which method should be used? so here is your answer, first method is separation of variables, this method is use when a differential equation can be written in the form dy/dx = f(y) g(x), where f is the function is y only and g is function of x only. To solve problems by this methods rewrite the problem as 1/f(y) dy= g(x) dx and integrate both side. Use the initial condition to find the constant of integration.

Let’s clear this method by one first order differential equation example. Solve dy/dx=xy with initial condition y(0)=1. So to solve this problem divide through by y we get 1/y dy/dx= x, now integrate both side with respect to x we get ∫ 1/y dy/dx= ∫ xdx, by solving we get ln(y)= x^2/2+c. now make y as the subject of equation so y= Ae^x^2/2 where A= e^c. this is the general solution of this equation.

Now our second method is Integrating factor this method is use when differential equation is give in the form dy/dx + p(x)y = q(x) where p and q are function of x only. Our next point is how do we find Integrating factor∫ The integrating factor is e^∫ p(x) dx. Now to solve problems by this method you need to rearrange the differential equation in to the standard form and find the integrating factor. Multiply through the integrating factor and rewrite the left hand side as the derivative of ye^ ∫ p(x) dx. Integrating both sides gives the general solution. For more information how to solve math problems visit math related expert online sites.

Now we will discus about what is first order nonlinear differential equation and what is first order ordinary differential equation. The first order differential equation is nonlinear if the function or its derivative is multiplied to itself.  One example is du/dx=u^2+1. The next is first order ordinary differential equation, in this differential equation the unknown function is a function of single independent variable.