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.

${\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{xy}\mathbf{\text{'}}\mathbf{+}\mathbf{12}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{}{\mathbf{x}}^{\mathbf{3}}\mathbf{,}{\mathbf{x}}^{\mathbf{4}}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Short answer

The given functions satisfy the given D.E and are linearly independently on the interval $(0,\infty ),andy=c_1{x}^3+c_2{x}^4$.

Step-by-step solution

Verify the function for the given differential equation

The third order differential equation

$x^{2}y\text{'}\text{'}-6xy\text{'}+12y=0$

With the functions$y_1=x^{3}and{y}_2=x^4$, and we have to verify the functions form a fundamental set of solutions of the given differential equation on the interval $(0,\infty )$as the following technique:

Verify the given functions satisfy the given differential equation as the following:

The first function:$y_1=x^3$

Differentiating the function with respect to x, then we have,

$y_1\text{'}=3x^2$

And

$y_1\text{'}\text{'}=6x$

Substitute the derivatives to the given differential equation. Then we have

$\begin{array}{rcl}{x}^2\times 6x-6x\times 3x^2+12x^3& =& 0\\ 6x^3-18x^3+12x^3& =& 0\\ 18x^3-18x^3& =& 0\\ 0& =& 0\end{array}$

The function $y_1=x^3$satisfies the given differential equation.

Solve the second function

The second function $y_2=x^4$:

Differentiating the function with respect to x, then we have,

$y_2\text{'}=4x^3$

And

$y_2\text{'}\text{'}=12x^2$

Substitute the derivatives to the given differential equation. Then we have

$\begin{array}{rcl}{x}^2\times 12x^2-6x\times 4x^3+12x^4& =& 0\\ 12x^4-24x^4+12x^4& =& 0\\ 24x^4-24x^4& =& 0\\ 0& =& 0\end{array}$

The function $y_2=x^4$satisfies the given differential equation.

We have to know the given function is linearly dependant or independent Using Wronskian as

$$\begin{gathered} W\left(y_1,y_2\right)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\text{'}}& y_2^{\text{'}}\end{array}\right| \\ =\left|\begin{array}{cc}{x}^3& x^4\\ 3x^2& 4x^3\end{array}\right| \end{gathered}$$

Determinate the matrix as

$\begin{array}{rcl}W\left(x^3,x^4\right)& =& x^3\times 4x^3-x^4\times 3x^2\\ & =& 4x^6-3x^6\\ & =& x^6\end{array}$

Since the determinate of the Wronskian of the given set of functions is not equal Zero, then this set of function is linearly independent,

The general solution is $y=c_1{x}^3+c_2{x}^4$.