Answer:
![x=-cos(t)+2sin(t)](https://tex.z-dn.net/?f=x%3D-cos%28t%29%2B2sin%28t%29)
Step-by-step explanation:
The problem is very simple, since they give us the solution from the start. However I will show you how they came to that solution:
A differential equation of the form:
![a_n y^n +a_n_-_1y^{n-1}+...+a_1y'+a_oy=0](https://tex.z-dn.net/?f=a_n%20y%5En%20%2Ba_n_-_1y%5E%7Bn-1%7D%2B...%2Ba_1y%27%2Ba_oy%3D0)
Will have a characteristic equation of the form:
![a_n r^n +a_n_-_1r^{n-1}+...+a_1r+a_o=0](https://tex.z-dn.net/?f=a_n%20r%5En%20%2Ba_n_-_1r%5E%7Bn-1%7D%2B...%2Ba_1r%2Ba_o%3D0)
Where solutions
are the roots from which the general solution can be found.
For real roots the solution is given by:
![y(t)=c_1e^{r_1t} +c_2e^{r_2t}](https://tex.z-dn.net/?f=y%28t%29%3Dc_1e%5E%7Br_1t%7D%20%2Bc_2e%5E%7Br_2t%7D)
For real repeated roots the solution is given by:
![y(t)=c_1e^{rt} +c_2te^{rt}](https://tex.z-dn.net/?f=y%28t%29%3Dc_1e%5E%7Brt%7D%20%2Bc_2te%5E%7Brt%7D)
For complex roots the solution is given by:
![y(t)=c_1e^{\lambda t} cos(\mu t)+c_2e^{\lambda t} sin(\mu t)](https://tex.z-dn.net/?f=y%28t%29%3Dc_1e%5E%7B%5Clambda%20t%7D%20cos%28%5Cmu%20t%29%2Bc_2e%5E%7B%5Clambda%20t%7D%20sin%28%5Cmu%20t%29)
Where:
![r_1_,_2=\lambda \pm \mu i](https://tex.z-dn.net/?f=r_1_%2C_2%3D%5Clambda%20%5Cpm%20%5Cmu%20i)
Let's find the solution for
using the previous information:
The characteristic equation is:
![r^{2} +1=0](https://tex.z-dn.net/?f=r%5E%7B2%7D%20%2B1%3D0)
So, the roots are given by:
![r_1_,_2=0\pm \sqrt{-1} =\pm i](https://tex.z-dn.net/?f=r_1_%2C_2%3D0%5Cpm%20%5Csqrt%7B-1%7D%20%3D%5Cpm%20i)
Therefore, the solution is:
![x(t)=c_1cos(t)+c_2sin(t)](https://tex.z-dn.net/?f=x%28t%29%3Dc_1cos%28t%29%2Bc_2sin%28t%29)
As you can see, is the same solution provided by the problem.
Moving on, let's find the derivative of
in order to find the constants
and
:
![x'(t)=-c_1sin(t)+c_2cos(t)](https://tex.z-dn.net/?f=x%27%28t%29%3D-c_1sin%28t%29%2Bc_2cos%28t%29)
Evaluating the initial conditions:
![x(0)=-1\\\\-1=c_1cos(0)+c_2sin(0)\\\\-1=c_1](https://tex.z-dn.net/?f=x%280%29%3D-1%5C%5C%5C%5C-1%3Dc_1cos%280%29%2Bc_2sin%280%29%5C%5C%5C%5C-1%3Dc_1)
And
![x'(0)=2\\\\2=-c_1sin(0)+c_2cos(0)\\\\2=c_2](https://tex.z-dn.net/?f=x%27%280%29%3D2%5C%5C%5C%5C2%3D-c_1sin%280%29%2Bc_2cos%280%29%5C%5C%5C%5C2%3Dc_2)
Now we have found the value of the constants, the solution of the second-order IVP is:
![x=-cos(t)+2sin(t)](https://tex.z-dn.net/?f=x%3D-cos%28t%29%2B2sin%28t%29)