djeljenike
2021-09-23
Answered

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.

$\lambda 1=-1\to \left\{\begin{array}{cc}1& 1\end{array}\right\},\lambda 2=2\to \left\{\begin{array}{cc}1& -1\end{array}\right\}$

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Aniqa O'Neill

Answered 2021-09-24
Author has **100** answers

asked 2021-06-10

Determine whether the given set S is a subspace of the vector space V.

A. V=${P}_{5}$ , and S is the subset of ${P}_{5}$ consisting of those polynomials satisfying p(1)>p(0).

B.$V={R}_{3}$ , and S is the set of vectors $({x}_{1},{x}_{2},{x}_{3})$ in V satisfying ${x}_{1}-6{x}_{2}+{x}_{3}=5$ .

C.$V={R}^{n}$ , and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=${C}^{2}(I)$ , and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=${P}_{n}$ , and S is the subset of ${P}_{n}$ consisting of those polynomials satisfying p(0)=0.

G.$V={M}_{n}(R)$ , and S is the subset of all symmetric matrices

A. V=

B.

C.

D. V=

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=

G.

asked 2022-05-18

I am studying about the linear odes with non-constant coefficients.

I know the first order linear ode with non-constant coefficient

$\begin{array}{}\text{(1)}& {y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

has a general solution of the form

$\begin{array}{}\text{(2)}& y=C{e}^{-\int f(x)dx}\end{array}$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$\begin{array}{}\text{(3)}& {y}^{{}^{\u2033}}(x)+g(x){y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.

Questions

1. Consider (3), when $g(x)=0$, then we have

$\begin{array}{}\text{(4)}& {y}^{{}^{\u2033}}(x)+f(x)y(x)=0\end{array}$

Is Eq.(4) a famous well-known equation? If YES, what is its name?

2. Does (4) have a closed form solution like (2)?

3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

I know the first order linear ode with non-constant coefficient

$\begin{array}{}\text{(1)}& {y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

has a general solution of the form

$\begin{array}{}\text{(2)}& y=C{e}^{-\int f(x)dx}\end{array}$

However, I am more interested in the case of linear second order odes with non-constant coefficients

$\begin{array}{}\text{(3)}& {y}^{{}^{\u2033}}(x)+g(x){y}^{{}^{\prime}}(x)+f(x)y(x)=0\end{array}$

I know that this equation does not have a closed form solution like (2). However, I am interested in special cases of that.

Questions

1. Consider (3), when $g(x)=0$, then we have

$\begin{array}{}\text{(4)}& {y}^{{}^{\u2033}}(x)+f(x)y(x)=0\end{array}$

Is Eq.(4) a famous well-known equation? If YES, what is its name?

2. Does (4) have a closed form solution like (2)?

3. Can you name or give me a list of well-known linear second order odes with non-constant coefficients which are not polynomial?

For example, I know Cauchy-Euler, Airy, Bessel, Chebyshev, Laguerre and Legendre equations whose coefficients are polynomials. But I don't know any well-known equation with non-polynomial coefficients.

asked 2021-09-25

-4x - 15y = -17

-x + 5y = -13

x = ___, y = ___

-x + 5y = -13

x = ___, y = ___

asked 2021-05-21

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named

asked 2021-06-24

Write the vector form of the general solution of the given system of linear equations.

asked 2021-09-29

Solve the system by clennaton

The solution is____

asked 2021-06-04

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution.