2. Consider the fourth-order boundary-value problem p''(x) – a*q(x) = 0 p(0) = p'(0) = P(1) = p"(1) = 0 where a > 0. Show that there is a non-trivial solution p(x) if and only if tan a = tanh a. Note: This boundary-value problem arises in modeling the vibrations of a beam that is clamped at one end and simply supported at the other. (See hint for problem 1.)

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.