Differential Equations True or False

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

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 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

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 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

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

True