Question

In Problems 23–30 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.

${\mathbf{y}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{1}\mathbf{,}\mathbf{x}\mathbf{,}\mathbf{cos}\mathbf{}\mathbf{x}\mathbf{,}\mathbf{sin}\mathbf{}\mathbf{x}\mathbf{,}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

Short answer

The function $y^{\left(4\right)}+y\text{'}\text{'}=0$ forms a fundamental set of solutions of the differential equation on the indicated interval $(-\infty ,\infty )$. The general solution is $y=c_1+c_{2}x+c_{3}cosx+c_{4}sinx$.

Step-by-step solution

Find the derivatives of the first and second functions

The first function is $y_1\left(x\right)=1$.

Determining the derivatives of the first function,

$$\begin{gathered} y\text{'}=0 \\ y\text{'}\text{'}=0 \\ {y}^{\left(3\right)}=0 \\ {y}^{\left(4\right)}=0 \end{gathered}$$

The second function is $y_2\left(x\right)=x$.

Determining the derivatives of the second function,

$$\begin{gathered} y\text{'}=1 \\ y\text{'}\text{'}=0 \\ {y}^{\left(3\right)}=0 \\ {y}^{\left(4\right)}=0 \end{gathered}$$

Verify the first two functions as solutions of the differential equation

Substituting the values of the derivatives of the first function in the differential equation,

$\begin{array}{rcl}{y}^{\left(4\right)}+y\text{'}\text{'}& =& 0+0\\ & =& 0\end{array}$

This is true.

Substituting the values of the derivatives of the second function in the differential equation,

$\begin{array}{rcl}{y}^{\left(4\right)}+y\text{'}\text{'}& =& 0+0\\ & =& 0\end{array}$

This is true.

Find the derivatives of the third and fourth functions

The third function is$y_3\left(x\right)=cosx$ .

Determining the derivatives of the third function,

$$\begin{gathered} y\text{'}=-sinx \\ y\text{'}\text{'}=-cosx \\ {y}^{\left(3\right)}=sinx \\ {y}^{\left(4\right)}=cosx \end{gathered}$$

The fourth function is$y_4\left(x\right)=sinx$ .

Determining the derivatives of the fourth function,

$$\begin{gathered} y\text{'}=cosx \\ y\text{'}\text{'}=-sinx \\ {y}^{\left(3\right)}=-cosx \\ {y}^{\left(4\right)}=sinx \end{gathered}$$

Verify the third and fourth functions as solutions of the differential equations

Substituting the values of the derivatives of the third function in the differential equation,

$\begin{array}{rcl}{y}^{\left(4\right)}+y\text{'}\text{'}& =& cosx+(-cosx)\\ & =& 0\end{array}$

This is true.

Substituting the values of the derivatives of the fourth function in the differential equation,

$\begin{array}{rcl}{y}^{\left(4\right)}+y\text{'}\text{'}& =& sinx+(-sinx)\\ & =& 0\end{array}$

This is true.

Verify the functions as solutions of the differential equations

Since all four functions satisfy the differential equation,

The functions given by:

$$\begin{gathered} y=1 \\ y=x \\ y=cosx \\ y=sinx \end{gathered}$$

All are the solutions of the differential equation.

Find the Wronskian of the function

Using the definition of the Wronskian of the function,

$\begin{array}{rcl}W(1,x,cosx,sinx)& =& \left|\begin{array}{cccc}1& x& cosx& sinx\\ 0& 1& -sinx& cosx\\ 0& 0& -cosx& -sinx\\ 0& 0& sinx& -cosx\end{array}\right|\\ & =& \left|\begin{array}{ccc}1& -sinx& cosx\\ 0& -cosx& -sinx\\ 0& sinx& -cosx\end{array}\right|\\ & =& (-cosx)(-cosx)-(-sinx)(sinx)\\ & =& co{s}^{2}x+si{n}^{2}x\\ & =& 1\end{array}$

Since the Wronskian is not zero for all values of x , all the functions are linearly independent on the interval $(-\infty ,\infty )$$(-\infty ,\infty )$.

Hence, from the definition of Fundamental Set of Solutions, the functions form a fundamental set of solutions of the differential equation on the interval $(-\infty ,\infty )$ .

Find the General Solution of the function

From the definition of general solution of homogeneous equations,

The required general solution is $y=c_1+c_{2}x+c_3\cos x+c_4\sin x$, where $c_1,c_2,c_3,c_4$ are arbitrary constants.