Differential Equations True or False

A second-order differential equation y'' = f(y', y, t) together with an initial condition y(t0) = y0 forms an Initial Value Problem.

False

The domain of the solution to a first-order differential equation y' = f(y, t) can be identified without solving the problem.

False

The functions f1(t), f2(t),...,fn(t) are said to be linearly independent on an interval I if there exists a set of constants k1, k2,...,kn, not all zero, such that k1f1(t) + k2f2(t) + ··· + knfn(t)=0 for all t in I. The functions f1(t),...,fn(t) are said to be linearly dependent on I if they are not linearly independent there.

False

A first-order differential equation y' = f(y, t) together with an initial condition y(t0) = y0 forms an Initial Value Problem.

True

A function f(x) that has a Taylor series expansion about x = x0 with a radius of convergence ⇢ > 0, is said to be analytic at x = x0.

True

Consider the problem y'' + p(t) y' + q(t) y = g(t). If yp(t) is a particular solution to the problem above and y1(t) and y2(t) are a fundamental set of solutions to the corresponding homogeneous problem, then the general solution to the listed problem is y(t) = y1(t) + y2(t) + yp(t).

True

For linear first-order differential equations it is always possible to write solutions explicitly

True

For linear second-order homogeneous differential equations with constant coefficients, it is always possible to write the general solution explicitly.

True

For the differential equation y'' + p(x) y' + q(x) y = 0: if the functions p(x) and q(x) are analytic at x0, then x0 is said to be an ordinary point of the differential equation; otherwise, it is a singular point.

True

Given the n-th order differential equation y(n) + p1(t) y(n1) + ... + pn1(t) y' + pn(t) y = 0, y1(t),...,yn(t) form a fundamental set of solutions of the given ODE on an interval I, if and only if they are linearly independent on I.

True

The first-order differential equation y' +p(t) y = g(t) is linear for any p(t) and g(t).

True

The general solution to a linear first-order differential equation contains one arbitrary constant.

True

The linear combination of the solutions y1(t) and y2(t) to y'' +p(t)y' + q(t)y = 0 contains all possible solutions to the problem if and only if the Wronskian, W(y1, y2)(t), is not everywhere zero.

True

The n-th order ODE y(n) + p1(t) y(n1) + ... + pn1(t) y' + pn(t) y = 0, is linear for any p1(t),...,pn(t)

True

The second-order ODE y'' + p(t) y' + q(t) y = g(t) is linear for any p(t), q(t), and g(t).

True