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## Glossary

# Boundary Value Problems

Consider a second order differential equation

**Theorem:** Suppose that *f(t,x,y)* is continuous on the region \( R = \left\{ (t,x,y)\, : \, a \le t \le b, \ -\infty < x < \infty , \ -\infty < y < \infty \right\} \) and that \( \partial f/\partial x = f_x \quad \partial f/\partial y = f_y \) are continuous on *R*. If there exists a positive constant *M* for which *f*_{x} and *f*_{y} satisfy

*x = x(t)*for \( a \le t \le b . \) ■

The notation \( y = x' (t) \) has been used to distinguish the third variable of the function \( f(t,x,x' ) . \) Recall that we also use the dot notation for derivatives with respect to time variable *t:* \( \dot{x} = x' (t) . \) Finally, the special case of linear differential equations is worthy of mention.

**Theorem:** Assume that *f* in the previous theorem has the linear form \( f(t,x,x' ) = p(t)\,x' + q(t)\,x + r(t) , \) and that *f* and its partial derivatives \( q(t) = \partial f/\partial x \) and *p(t)* are continuous on *R*. If there exists a positive constant *M* for which *p(t)* and *q(t)* satisfy

*x = x(t)*for \( a \le t \le b . \) ■

Conditions of the above theorem are fulfilled for constant coefficient equations when *p(t) = p* and *q(t) = q*. In this case, a closed form formula for solution is possible to obtain.

yb[x_] = y[x] /. DSolve[L[x, y] == 36 x, y[x], x][[1]]

const = Solve[{yb[0] == 2, yb[1] == 1, yb[2] == -1}, {C[1], C[2], C[3]}]

z[x_] = Simplify[yb[x] /. const[[1]]]

E^2 + E^3 + E^4)) + ( E^(-1 + x) (-10 + 18 E^2 + 3 E^3 - 10 E^5))/((-1 + E)^2 (1 + E) (1 +

E + E^2)) + (E^(4 - 2 x) (-18 + 10 E + 10 E^3 - 3 E^4))/(

1 - E^3 - E^5 + E^8) + 6 x

zb[x_] = yb[x] /. {C[1] -> p1, C[2] -> p2, C[3] -> p3}

Plot[zb[x], {x, -1, 2.5}]

- Agarwal, R.P., Sheng, Q., Wong, P.J.Y., Abel-Gontscharoff boundary value problems, Math Comput Modeling, 1993, Vol. 17, No. 7, pp. 37--55.

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