Differential equations

Exercise 1.1

1.. Show that $e^{x},xe^{x},c_{1}e^{x}+c_{2}xe^{x}$ are solutions of $y^{\prime\prime} -2y^{\prime} + y = 0$ .

For $y(0) = 1,\ y^{\prime}(0) = -1$ ，find $c_{1},c_{2}$ .
2.. Show that $\sin{x},\cos{x}$ and their linear combination is the solution of $y^{\prime\prime} + y = 0$ . What can you say about the spanning vector space by $\sin{x}$ and $\cos{x}$ ?

Answer
1. Since $y = e^{x}$ , we have $y^{\prime} = e^{x}, y^{\prime\prime} = e^{x}$ .Thus,

$\displaystyle y^{\prime\prime} -2y^{\prime} + y = e^{x} - 2e^{x} + e^{x} = 0$

For $y = xe^{x}$ , we have $y^{\prime} = e^{x} + xe^{x}, y^{\prime\prime} = 2e^{x} + xe^{x}$ . Thus,

$\displaystyle y^{\prime\prime} -2y^{\prime} + y = 2e^{x} + xe^{x} - 2(e^{x} + xe^{x}) + xe^{x} = 0$

For $y = c_{1}e^{x}+c_{2}xe^{x}$ , we have $y^{\prime} = c_{1}e^{x} + c_{2}(e^{x} + xe^{x}), y^{\prime\prime} = c_{1}e^{x} + c_{2}(2e^{x} + xe^{x})$ . Thus,

$\displaystyle y^{\prime\prime} -2y^{\prime} + y$	$\displaystyle =$	$\displaystyle c_{1}e^{x} + c_{2}(2e^{x} + xe^{x}) - 2(c_{1}e^{x} + c_{2}(e^{x} + xe^{x})) + c_{1}e^{x}+c_{2}xe^{x}$
	$\displaystyle =$	$\displaystyle c_{1}(e^{x} - 2e^{x} + e^{x}) + c_{2}(2e^{x} + xe^{x} - 2(e^{x} + xe^{x}) + xe^{x}) = 0$

For $y = c_{1}e^{x}+c_{2}xe^{x}, \ y(0) = 1,\ y^{\prime}(0) = -1$ , we find $c_{1},c_{2}$ . Since $y(0) = 1$

, $1 = y(0) = c_{1}$ .Thus, $c_{1} = 1$ . $y^{\prime}(0) = -1$ implies that $-1 = y^{\prime}(0) = c_{1} + c_{2}$ . Then $c_{2} = -2$ .

2. For $y = \sin{x}$ , $y^{\prime} = \cos{x}, y^{\prime\prime} = -\sin{x}$ . Thus,

$\displaystyle y^{\prime\prime} + y = -\sin{x} + \sin{x} = 0$

For $y = \cos{x}$ , $y^{\prime} = -\sin{x}, y^{\prime\prime} = -\cos{x}$ . Thus,

$\displaystyle y^{\prime\prime} + y = -\cos{x} + \cos{x} = 0$

For $y = c_{1}\sin{x} + c_{2}\cos{x}$ , $y^{\prime} = c_{1}\cos{x} - c_{2}\sin{x}, y^{\prime\prime} = -c_{1}\sin{x} - c_{2}\cos{x}$ . Thus

$\displaystyle y^{\prime\prime} + y = -c_{1}\sin{x} - c_{2}\cos{x} + c_{1}\sin{x} + c_{2}\cos{x} = 0$

We consider the vector space spanned by $\sin{x}$ and $\cos{x}$ . Note that the vector space spanned by $\sin{x}$ ans $\cos{x}$ can be expressed by the following.

$\displaystyle V(\sin{x},\cos{x}) = \{c_{1}\sin{x} + c_{2}\cos{x} : c_{1},c_{2} \in R\}$

In other words, it is a linear combination of $\sin{x}$ and $\cos{x}$ ，We now find the dimension of this vector space.

$\displaystyle c_{1}\sin{x} + c_{2}\cos{x} = 0, c_{1}\cos{x} - c_{2}\sin{x} = 0$

implies that $c_{1} = c_{2} = 0$ . Thus, we can say that $\sin{x}$ and $\cos{x}$ are linearly independent. Therefore，the dimension of $V(\sin{x},\cos{x})$ is 2.