The picture below (click on it to enlarge) and at the right shows a particular solution corresponding to y(0) = y_o = 1.  Click here or click on the picture at the right to see an animation (y_o varying from -1/2 to -5 (blue) and y_o varying from -1/2 to 4 (red) ).  Click here to see a similar animation that includes negative values for x.

Here is a DPGraph picture of z = -ye^-x + cos(x) - y²

and z = C.  You can use the scrollbar to vary C.  The intersection of the two surfaces would correspond to the solution for each value of C.  You can also use the z-slice feature to look at various solutions corresponding to z = C.

Here is a DPGraph picture of

z = -ye^-x + cos(x) - y² - 1 + a + a²  where a = y_o.

C = 1 - a - a²

The view is from the top with z between -0.0001 and +0.0001 so you have the effect of looking at the graph of

-ye^-x + cos(x) - y² - 1 + a + a² = 0

You can use the scrollbar to vary a (y_o) from -4 to 5.  The default value for a is 1.

Closer look at (0,-1/2)

Quicktime version of the animation

Try using the First Order DE Solution Grapher to solve the DE in standard form and observe what happens when you reach a value for x where the analytical solution is undefined.

                                        Integrating Factor Formulas

The three pictures on the right show particular solutions to the equation above with y(0) = y_o = -3, 1, and 3.  EC:  What is not clearly shown in the pictures where y_o = -3 and 3?  (Hint:  Your graphing calculator does the same thing.)   Click here to see an animation of particular solutions with y_o varying from -5 to 5. Quicktime version