On the solution spaces of linear second-order homogeneous ordinary differential equations and associated boundary conditions
There are given a lower bound \(h(t)\) and an upper bound \(H(t)\) for the solution \(x(t)\) of the initial value problem (1) \(x'''+ a_1(t) x''+ a_2(t) x'+ a_3(t) x= 0\), (2) \(x(t_0)= x_0,\;x'(t_0)= x_1,\;x''(t_0)= x_2\), on an interval \([t_0, t_1]\) under the assumption that there exists a positive solution \(p(t)\) of (1) on \([t_0, t_1]\). To obtain the bounds \(h(t)\) and \(H(t)\) the author uses the following procedure. By the substitution \[ x(t)= p(t)\Biggl( {x_0\over p(t_0)}+ \int^t_{t_0} w(s) ds\Biggr),\quad w\in C^2([t_0, t_1]), \] the problem (1), (2) is transformed to an initial value problem for a second-order linear differential equation. Let \(y(t)\) be the solution of the last initial value problem and let \(h_1(t)< y(t)< H_1(t)\) on \([t_0, t_1]\) with continuous functions \(h_1(t)\), \(H_1(t)\). Then \(h(t)\) (resp. \(H(t)\)) is expressed by \(p(t)\) and \(h_1(t)\) (resp. \(p(t)\) and \(H_1(t)\)). Some examples are given.
second-order linear differential equation
scientific article; zbMATH DE number 894217