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 single linear n th order equation. the solutions will...

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 single linear n th order equation. the solutions will be x(t) and y(t). Need fully analytical work.

b)   If t = [ 0, 2`pi`/10], sketch and describe the shape of the curve represented by the parametric equation given by x(t) and y(t) and also show the location of your initial condition x(0) and y(0). You can use parametric plots in matlab or just sketch by hand.  

 

Please provide as many steps as possible if you can't get the final answer.

 

Expert Answers
Neethu eNotes educator| Certified Educator

(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.

 

 

 

 

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