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:

`x'=10y`

`y'=-10x`

with initial conditions:

`x(0)=3`

`y(0)=4`

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

`[[x'],[y']]=[[0,10],[-10,0]][[x],[y]]`

So let `A=[[0,10],[-10,0]]`

Now let us write the characteristic equation i.e.

`|A-lambda I |=0`

`|[-lambda,10],[-10,-lambda]|=0`

`lambda^2+100=0`

`rArr lambda = +-10i`

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

For `lambda_1=10i`

We have,

`[[-10i,10],[-10,-10i]][[v_1],[v_2]]=[[0],[0]]`

i.e.

...

(a)  

We have to solve the initial value problem given by:

`x'=10y`

`y'=-10x`

with initial conditions:

`x(0)=3`

`y(0)=4`

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

`[[x'],[y']]=[[0,10],[-10,0]][[x],[y]]`

So let `A=[[0,10],[-10,0]]`

Now let us write the characteristic equation i.e.

`|A-lambda I |=0`

`|[-lambda,10],[-10,-lambda]|=0`

`lambda^2+100=0`

`rArr lambda = +-10i`

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

For `lambda_1=10i`

We have,

`[[-10i,10],[-10,-10i]][[v_1],[v_2]]=[[0],[0]]`

i.e.

`-10i v_1+10v_2=0`

`-iv_1+v_2=0 rArr v_2=iv_1`

or,

`-10v_1-10iv_2=0`

`v_1+iv_2=0`

i.e. `-iv_1+v_2=0 rArr v_2=iv_1`

So we have the eigen vector as:

`eta_1=[[v_1],[v_2]]=[[1],[i]]` `=[[1],[0]]+[[0],[1]]i`

when `v_1=1`

So now we can write the solution as:

Since we have complex conjugate eigen values of the form `mu+-lambda i` and suppose `eta = a+bi ` is the eigen vector,

our solution will be of the form:

`[[x],[y]]=C_1 e^{mu t}(a cos(lambda t)-bsin( lambda t))+C_2e^{mu t}(asin(lambda t)+bcos(lambda t))`

i.e.  `[[x],[y]]=C_1e^{0 t}([[1],[0]]cos(10 t)-[[0],[1]]sin(10t))+C_2e^{0 t}([[1],[0]]sin(10 t)+[[0],[1]]cos(10 t))`

           `=C_1[[cos(10t)],[-sin(10t)]]+C_2[[sin(10t)],[cos(10t)]]`

Now applying the initial conditions we have,

`[[x(0)],[y(0)]]=[[3],[4]]=C_1[[1],[0]]+C_2[[0],[1]]`

i.e.

`C_1=3` and  `C_2=4`

Hence we have the final solution as:

`[[x(t)],[y(t)]]=3[[cos(10t)],[-sin(10t)]]+4[[sin(10t)],[cos(10t)]]`

i.e.

`x(t)=3cos(10t)+4sin(10t)`  and,

`y(t)=-3sin(10t)+4cos(10t)`

(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.