Question

Using the Wronskian in this Problem, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.

$$\begin{gathered} {\mathbf{x}}^{\mathbf{3}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{3}{\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{xy}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{>}\mathbf{0}\mathbf{;} \\ \left\{x, x^2, x^3\right\} \end{gathered}$$

Short answer

Thus, it is verified that the given functions form a fundamental solution set for the given differential equation, and therefore, the general solution is$\mathbf{y}\mathbf{=}\mathbf{Ax}\mathbf{+}{\mathbf{Bx}}^{\mathbf{2}}\mathbf{+}{\mathbf{Cx}}^{\mathbf{3}}.$

Step-by-step solution

Step 1:Using the concept of Wronskian

The given function is$\left\{\mathbf{x}\mathbf{,} {\mathbf{x}}^{\mathbf{2}}\mathbf{,} {\mathbf{x}}^{\mathbf{3}}\right\}.$

Apply the concept of Wronskian,

$W\left[f_1,f_2,\dots ,f_n\right]=\left|\begin{array}{cccc}{f}_1\left(x\right)& f_2\left(x\right)& \dots & f_n\left(x\right)\\ f_1\text{'}\left(x\right)& f_2\text{'}\left(x\right)& \dots & f_n\text{'}\left(x\right)\\ ⋮& ⋮& & ⋮\\ {f_1}^{n-1}\left(x\right)& {f_2}^{n-1}\left(x\right)& \cdots & {f_n}^{n-1}\left(x\right)\end{array}\right|$

Therefore,

$W\left[x,x^2,x^3\right]=\left|\begin{array}{ccc}x& x^2& x^3\\ 1& 2x& 3x^2\\ 0& 2& 6x\end{array}\right|$

Solve the above equation,

$$\begin{gathered} W\left[x,x^2,x^3\right]=\left|\begin{array}{ccc}x& x^2& x^3\\ 1& 2x& 3x^2\\ 0& 2& 6x\end{array}\right| \\ =x\left(12x^2-6x^2\right)-x^2\left(6x\right)+x^3\left(2\right) \\ =6x^3-6x^3+2x^3 \\ =2x^3 \end{gathered}$$

Step 2:Find a general solution

The Wronskian of the above functionis never zero on the interval $\left(0,\infty \right)$.

Thus, it isverified that the given functions form a fundamental solution set for the given differential equation.

Therefore, the general solution is$\mathbf{y}\mathbf{=}\mathbf{Ax}\mathbf{+}{\mathbf{Bx}}^{\mathbf{2}}\mathbf{+}{\mathbf{Cx}}^{\mathbf{3}}.$