# Linear equations of second order with constant coefficients. Find all solutions on {left(-infty,+inftyright)}.{y}text{}+{2}{y}'+{y}={0}

Question
Linear equations of second order with constant coefficients.
Find all solutions on
$${\left(-\infty,+\infty\right)}.{y}\text{}+{2}{y}'+{y}={0}$$

2020-12-25
The characteristic equation is given by
$${r}^{2}+{2}{r}+{1}={0}$$
and has the roots $$r_1=11 and r_2=1$$.
The general solution is given by
$${y}={e}^{x}{\left({c}_{{1}}+{c}_{{2}}{x}\right)}$$, where c_1 and c_2 are arbitracy constant.

### Relevant Questions

Linear equations of second order with constant coefficients.
Find all solutions on
$$\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}-{2}{y}'+{5}{y}={0}$$
Linear equations of second order with constant coefficients. Find all solutions on $$\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}+{4}{y}={0}$$
$$y=3e^{3x}$$ is a solution of a second order linear homogeneous differential equation with constant coefficients. The equation is:
(a) $$y''-(3+a)y'+3ay=0$$, a any real number.
(b) $$y''+y'-6y=0$$
(c) $$y''+3y'=0$$
(d) $$y''+(3-a)y'+3ay=0$$, a any real number.
(e) Cannot be determined.
Linear equations of first order.
Solve the initial-value problem on the specified interval $$\displaystyle{y}'-{3}{y}={e}^{{{2}{x}}}{o}{n}{\left(-\infty,+\infty\right)},\ \text{with}\ {y}={0}\ \text{when}\ {x}={0}$$.
Consider the differential equation for a function f(t),
$$tf"(t)+f'(t)-f((t))^2=0$$
a) What is the order of this differential equation?
b) Show that $$f(t)=\frac{1}{t}$$ is a particular solution to this differential equation.
c)Find a particular solution with f(0)=0
2. Find the particular solutions to the differential equations with initial conditions:
a)$$\frac{dy}{dx}=\frac{\ln(x)}{y}$$ with y(1)=2
b)$$\frac{dy}{dx}=e^{4x-y}$$ with y(0)=0
Solve the second order linear differential equation using method of undetermined coefficients
$$3y''+2y'-y=x^2+1$$
$$t(t^2-4)y''-ty'+3t^2y=0, y(1)=1 y'(1)=3$$
$$y''-4y'+9y=0,\ \ y(0)=0,\ \ y'(0)=-8$$
Let $$y_1$$ and $$y_2$$ be solution of a second order homogeneous linear differential equation $$y''+p(x)y'+q(x)=0$$, in R. Suppose that $$y_1(x)+y_2(x)=e^{-x}$$,
$$W[y_1(x),y_2(x)]=e^x$$, where $$W[y_1,y_2]$$ is the Wro
ian of $$y_1$$ and $$y_2$$.
Find p(x), q(x) and the general form of $$y_1$$ and $$y_2$$.
On solution of the differetial equation $$y''+y'=0$$ is $$y=e^{-x}$$. Use Reduction of Order to find a second linearly independent solution.