- #1

psie

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- Homework Statement
- Find a fundamental solution of the equation ##x''-x=0##. Solve using this the equation ##x''-x=e^t\sin{t}##.

- Relevant Equations
- The definition of a fundamental solution is given in theorem below.

I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers.

Now, consider the equation ##x''-x=e^t\sin{t}##. The homogeneous solution is ##x_h(t)=Ae^t+Be^{-t}##. To obtain ##E(t,\tau)##, note that ##x_h'(t)=Ae^t-Be^{-t}##. According to ##(3)##, ##x_h(\tau)=x_h'(\tau)=0##, so we have two equations and two unknowns. Solving this system, we obtain ##A=\frac{e^{-\tau}}{2}## and ##B=-\frac{e^\tau}{2}##, so $$E(t,\tau)=\frac12(e^{t-\tau}-e^{\tau-t})=\sinh{(t-\tau)}.$$

Here is where I'm stuck. ##(4)## seems to depend on an initial value ##t_0##, which I don't have. Moreover, I'm a little unsure whether or not ##(4)## is the general solution to ##L(t,D)y=g(t)## or simply a particular solution. I'm looking to obtain the general solution of ##x''-x=e^t\sin{t}## using Green's function, i.e. ##E(t,\tau)##. Grateful for any help.

##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also known as Green's function.Theorem.Let $$L(t,\lambda)=\lambda^n+a_{n-1}(t)\lambda^{n-1}+\ldots+a_1(t)\lambda+a_0(t)\quad\text{and }D=\frac{d}{dt}.\tag1$$ Denote by ##E(t,\tau)## the uniquely determined solution ##u(t)## of the initial value problem

\begin{align}

&L(t,D)u=0 \tag2\\

&u(\tau)=u'(\tau)=\ldots=u^{(n-2)}(\tau)=0,\quad u^{(n-1)}(\tau)=1. \tag3

\end{align}

Then

$$y(t)=\int_{t_0}^t E(t,\tau)g(\tau)d\tau\tag4$$

is the solution of the problem

\begin{align}

&L(t,D)y=g(t) \tag5\\

&y(t_0)=y'(t_0)=\ldots=y^{(n-1)}(t_0)=0. \tag6

\end{align}

Now, consider the equation ##x''-x=e^t\sin{t}##. The homogeneous solution is ##x_h(t)=Ae^t+Be^{-t}##. To obtain ##E(t,\tau)##, note that ##x_h'(t)=Ae^t-Be^{-t}##. According to ##(3)##, ##x_h(\tau)=x_h'(\tau)=0##, so we have two equations and two unknowns. Solving this system, we obtain ##A=\frac{e^{-\tau}}{2}## and ##B=-\frac{e^\tau}{2}##, so $$E(t,\tau)=\frac12(e^{t-\tau}-e^{\tau-t})=\sinh{(t-\tau)}.$$

Here is where I'm stuck. ##(4)## seems to depend on an initial value ##t_0##, which I don't have. Moreover, I'm a little unsure whether or not ##(4)## is the general solution to ##L(t,D)y=g(t)## or simply a particular solution. I'm looking to obtain the general solution of ##x''-x=e^t\sin{t}## using Green's function, i.e. ##E(t,\tau)##. Grateful for any help.