The differential equation of the first order belongs to the simplest differential equations. They most easily give in to a research and the decision, and finally they can always be integrated.
1. We will consider the solution of the differential equation of the first order on the example of xy '=y. You see that it contains: x - an independent variable; at - a dependent variable, function; y' - the first derivative of function. Be not frightened if in certain cases in the equation the first order is not "x" or (and) "at". The main thing that in the differential equation surely there was y' (the first derivative), and there were no y'', y ''' (derivatives of the highest orders).
2. Present a derivative in the following form: y '=dydx (the formula is familiar from the school program). Your derivative has to look as follows: x*dydx=y, where dy, dx - differentials.
3. Now divide variables. For example, in the left part leave only the variable containing y, and in right - the variables containing x. At you the following has to turn out: dyy=dxx.
4. Integrate the differential equation received in the previous manipulations. So that's that: dyy=dxx
5. Now calculate the available integrals. In this simple case they are tabular. You have to receive the following result: lny=lnx+CEsli your answer differs from presented here, check all records. Somewhere the mistake is made and it needs to be corrected.
6. After integrals are calculated, the equation can be considered solved. But the received answer is submitted implicitly. On this step you received the general integral. submit to lny=lnx+CTeper the answer in an explicit form or, in other words, to find the common decision. Rewrite the answer received on the previous step in the following look: lny=lnx+C, use one of properties of logarithms: and from here express lna+lnb=lnab for the right member of equation (lnx+C) at. You have to receive record: lny=lnCx
7. Now remove logarithms and modules from both parts: y=Cx, With – consvy you have the function presented in an explicit form. It is also called the common decision for the differential equation of the first order of xy '=y.