# Initial value problem general form

Solve the initial value problem:

$$\frac{du}{dt}= \pmatrix{1&2\\-1&1}u, u(0) = \pmatrix{1\\0}$$

What I got is,

Eigenvalues : $\lambda = 1 \pm \sqrt{2}i$

Eigenvectors : $v_1 = \pmatrix{1\\-\frac{1}{\sqrt {2}i}},v_2 = \pmatrix{1\\\frac{1}{\sqrt {2}i}}$

Then, $$[e^tcos\sqrt{2}t + ie^tsin\sqrt{2}t]\pmatrix{1\\-\frac{1}{\sqrt {2}i}}$$ the problem I am having is the initial conditions. Can anyone show me how to do it?

The answer to this problem is

$$u(t) = \pmatrix{e^tcos\sqrt{2}t\\-\frac{1}{\sqrt {2}}e^tsin\sqrt{2}t}$$ I do not know how they got that.

-

Joriki, I dont understand what you mean by singular. Also, I thought that if you have complex conjugates eigenvalues then you dont have to write both the eigenvectors and just use the equation $[e^vtcosut + ie^vtsinut]$, where $\lambda = v \pm ui$ – Q.matin Nov 29 '12 at 5:49