The graph of a solution of the differential equation

intersects the graph of a solution of the differential equation

at the origin. Determine formulas for the functions and if the curves have equal slopes at the origin and if

First, we compute the general solutions of the two second-order differential equations.

is of the form

This gives us so and . Hence, the general solutions are

The other equation

is of the form

Therefore, so and the general solutions are given by

Letting and denote the particular solutions we are looking for (and renaming the constants and in the equations for and ) we have

We are given and . So,

and

Finally, we are also given

Putting these all together and solving for the constants we obtain

Therefore, the functions we are looking for are

Did you use matrices to solve for the constants? I’m not sure how to solve for them. Thanks.