Exact Solution of 1^st Order ODE with Fixed Constants: Example

In this segment, we will find the exact solution of a first order ordinary differential equation. So, let's go and take this example here. We are asked to solve this first order ordinary differential equation. This is the initial condition of that ordinary differential equation and we're supposed to find y at three. Solve is of this form so, let's go and see that how can we solve this particular problem. So, the solution will be the homogenous part plus the particular part so, let's concentrate on the homogeneous part.

So, in order to find out the homogeneous part of the solution what we're going to do is we're going to say that hey, let's define our differential operator to be d/dx like this. Then this differential equation can be written like this, then we get d + 0.4 being the operator for y. And the reason why we wrote it like this because this then directly gives us the characteristic equation, which will be m + 0.4 = 0. That will give us m is equal to -0.4, then that’ll give us the homogenous part of solution is k*e to the power m*x, so that is e to the power -0.4x. That is the homogeneous part of the solution.

Now let's go and find the particular part. The particular part will be of the form a*e to the power -x, why is that so? Because if you look at the right hand side the forcing function of this differential equation it is 3e to the power -x so, all its derivatives and its form is of the form of e to the power -x because the derivative of e to the power -x is -e to the power -x derivative of that is +e to the power -x so you always getting the same form of e to the power minus x. So that can be all bundled into just one single expression of a*e to the power -x. How do we go about finding what a is now? We substitute it back into our differential equation, the particular part of the solution which we just have given the form of, and let's see what happens. So, we take d/dx of e to the power -x here so. Derivative of this is -a to the power -x and you get this quantity right here. And this will be 0.6 a*e to the power -x, and this here gives you a equal to -5. What this means is that the particular part of the solution is nothing but minus -5e to the power -5. So, we’ve found the particular part of solution. So, let's go and now add the homogeneous part in the particular part to find out what the complete solution is.

So, the complete solution would be the homogeneous part plus the particular part. The homogeneous part is k*e to the power -0.4x. The particular part is -5 e to the power -x. So, how do we find k? We find k by knowing that the initial condition is given as 5. So, substituting the value of x equal to zero into this gives us five equal to k to the power minus this quantity minus five e to the power minus zero here. And that's k minus five that gives us k is equal to 10. So, what that means is that the whole solution will be 10 minus 5 e to the power -0.4x minus 5*e to the power -x.

So, that is the solution to the ordinary differential equation, but we are asked to find out what the value of y at 3 is, which will be this quantity right here. And this number here turns out to be 2.7630. So, that is the exact value of y(3) up to five significant digits based on the exact solution, and that is the end of this segment.