a) I need help to solve the initial value problem given by x' = 10y, y' = -10x, x(0) = 3 and y(0) = 4, by converting the system into a...

(a)  

We have to solve the initial value problem given by:

with initial conditions:

Now we can write the above problem in matrix form as shown:

So let

Now let us write the characteristic equation i.e.

Now we have to find the eigen vectors corresponding to the one of the eigen values  obtained above.

For

We have,

i.e.

...

(a)  

We have to solve the initial value problem given by:

with initial conditions:

Now we can write the above problem in matrix form as shown:

So let

Now let us write the characteristic equation i.e.

Now we have to find the eigen vectors corresponding to the one of the eigen values  obtained above.

For

We have,

i.e.

or,

i.e.

So we have the eigen vector as:

when

So now we can write the solution as:

Since we have complex conjugate eigen values of the form and suppose is the eigen vector,

our solution will be of the form:

i.e. 

          

Now applying the initial conditions we have,

i.e.

and 

Hence we have the final solution as:

i.e.

  and,

(b) 

Now we will sketch the graphs of the parametric equations x(t) and y(t)

The graph is of the shape of a circle with radius 5.

Location of initial conditions are also shown.