Plotting solution DSolve

Question

I am trying to plot a solution of a differential equation system. This is what I have:

eqn = {x''[t] + w^2*x[t] == 0, y[t] == x'[t]}; w = 1;
sol = DSolve[eqn, {x[t], y[t]}, t] 

and i need to plot 3 graphs x[t], y[t] and y[x]. I learned how to build the first two,

Plot[Evaluate[x[t] /. sol /. {C[1] -> 1, C[2] -> 1}], {t, 0, 10}]
Plot[Evaluate[y[t] /. sol /. {C[1] -> 1, C[2] -> 1}], {t, 0, 10}]

and i dont know how to build last y[x] and how to understand this. I add photo my task.

Answer 1

This is more of a Mathematics question.

eqn = {x''[t] + w^2*x[t] == 0, y[t] == x'[t]}; w = 1;
sol = DSolve[eqn, {x[t], y[t]}, t];
xt = FullSimplify[x[t] /. sol /. {C[1] -> 1, C[2] -> 1}][[1]]

$$x(t)=-\frac{\sin(t)}{2}$$

yt = FullSimplify[y[t] /. sol /. {C[1] -> 1, C[2] -> 1}][[1]]

$$y(t)=-\frac{\cos(t)}{2}$$

Now, recalling the trignometric identity and using the appropriate forms of the above two solutions,

$$1=\cos^2(t)+\sin^2(t)=4y^2+4x^2,$$

which gives equation of circle $x^2+y^2=1/4$ with radius $1/2$. Thus, your $y=f(x)$ is

$$y=\pm\sqrt{\left(\frac{1}{2}\right)^2-x^2}$$.

Plot[{Sqrt[(1/2)^2 - x^2], -Sqrt[(1/2)^2 - x^2]}, {x, -0.5, 0.5}, AspectRatio -> Automatic]

As I suggested in comment, the appropriate command to use is ParametricPlot.

In a parametric plot, you give both the $x$ and $y$ coordinates of each point as a function of a third parameter, say $t$.

Plotting (drawing) the circle with radius $1/2$ combine with parametric plot, we will see that both are the same,

Show[ParametricPlot[ {xt, yt}, {t, 0, 10}], ContourPlot[(x)^2 + (y)^2 == 1/4, {x, -5, 5}, 
                    {y, -3, 3}, ContourStyle -> {Red, Directive[Dashed]}]], Frame -> True]