Q15E - Fundamentals Of Differential Equations And Boundary Value Problems

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.

$$\begin{gathered} \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{11}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{12}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \\ \left\{e^{3x}, e^{-x}, e^{-4x}\right\} \end{gathered}$$

Step-by-Step Solution

Answer

Thus, it is verified that the given functions form a fundamental solution set for the given differential equation, and the general solution is $\mathbf{y}\mathbf{=}{\mathbf{Ae}}^{\mathbf{3}\mathbf{x}}\mathbf{+}{\mathbf{Be}}^{\mathbf{-}\mathbf{x}}\mathbf{+}{\mathbf{Ce}}^{\mathbf{-}\mathbf{4}\mathbf{x}}.$

Step 1: Using the concept of Wronskian

The given function is $\left\{{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{4}\mathbf{x}}\right\}$

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,

$W\left[e^{3x},e^{-x},e^{-4x}\right]=\left|\begin{array}{ccc}{e}^{3x}& e^{-x}& e^{-4x}\\ 3e^{3x}& -e^{-x}& -4e^{-4x}\\ 9e^{3x}& e^{-x}& 16e^{-4x}\end{array}\right|$

Solve the above equation,

$$\begin{gathered} W\left[e^{3x},e^{-x},e^{-4x}\right]=\left|\begin{array}{ccc}{e}^{3x}& e^{-x}& e^{-4x}\\ 3e^{3x}& -e^{-x}& -4e^{-4x}\\ 9e^{3x}& e^{-x}& 16e^{-4x}\end{array}\right| \\ \mathbf{=}{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\left(\mathbf{-}\mathbf{16}{\mathbf{e}}^{\mathbf{-}\mathbf{5}\mathbf{x}}\mathbf{+}\mathbf{4}{\mathbf{e}}^{\mathbf{-}\mathbf{5}\mathbf{x}}\right)\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\left(\mathbf{48}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{+}\mathbf{36}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right)\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{4}\mathbf{x}}\left(\mathbf{3}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{+}\mathbf{9}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\right) \\ \mathbf{=}{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\left(\mathbf{-}\mathbf{12}{\mathbf{e}}^{\mathbf{-}\mathbf{5}\mathbf{x}}\right)\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\left(\mathbf{84}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right)\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{4}\mathbf{x}}\left(\mathbf{12}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\right) \\ \mathbf{=}\mathbf{-}\mathbf{12}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{-}\mathbf{84}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{+}\mathbf{12}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}} \\ \mathbf{=}\mathbf{-}\mathbf{84}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}} \end{gathered}$$

Step 2: Find a general solution

The Wronskian of the above function is never zero on the interval $\left(a,b\right)$.

 

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

 

Therefore, the general solution is $\mathbf{y}\mathbf{=}{\mathbf{Ae}}^{\mathbf{3}\mathbf{x}}\mathbf{+}{\mathbf{Be}}^{\mathbf{-}\mathbf{x}}\mathbf{+}{\mathbf{Ce}}^{\mathbf{-}\mathbf{4}\mathbf{x}}.$