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# Determine functions u and v satisfying given differential equations

The graph of a solution of the differential equation

intersects the graph of a solution of the differential equation

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

First, we compute the general solutions of the second-order differential equation

This is an equation of the form

From this we compute , so and . Using Theorem 8.7 (pages 326-327 of Apostol) we have

Next, we compute the general solutions of the second-order differential equation

This is again an equation of the form

From this we compute so and . Again, applying Theorem 8.7 (pages 326-327 of Apostol) we have

Now, let

We know that the solutions we want intersect at the origin, hence we must have . So,

Plugging these values for the constants and into our expressions for and we compute the first derivatives,

Next, we are given that the particular solutions and we want have equal slopes at the origin. This means . Hence,

Finally,

Thus,