Q18E - 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.

$$\begin{gathered} {\mathbf{y}}^{\mathbf{4}}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \\ \left\{e^x, e^{-x}, \mathrm{cosx}, \mathrm{sinx}\right\} \end{gathered}$$

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 $\mathbf{y}\mathbf{=}{\mathbf{Ae}}^{\mathbf{x}}\mathbf{+}{\mathbf{Be}}^{\mathbf{-}\mathbf{x}}\mathbf{+}\mathbf{Ccosx}\mathbf{+}\mathbf{Dsinx}.$

Step 1: Using the concept of Wronskian

The given function is $\left\{{\mathbf{e}}^{\mathbf{x}}\mathbf{,} {\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{,} \mathbf{cosx}\mathbf{,} \mathbf{sinx}\right\}.$

 

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, the Wronskian of the given function is given as;

 

$W\left[e^x,e^{-x},\cos x,\sin x\right]=\left|\begin{array}{cccc}{e}^x& e^{-x}& \cos x& \sin x\\ e^x& -e^{-x}& -\sin x& \cos x\\ e^x& e^{-x}& -\cos x& -\sin x\\ e^x& -e^{-x}& \sin x& -\cos x\end{array}\right|$

 

Solve the above equation,

$$\begin{gathered} W\left[e^x,e^{-x},\cos x,\sin x\right]=\left|\begin{array}{cccc}{e}^x& e^{-x}& \cos x& \sin x\\ e^x& -e^{-x}& -\sin x& \cos x\\ e^x& e^{-x}& -\cos x& -\sin x\\ e^x& -e^{-x}& \sin x& -\cos x\end{array}\right| \\ =e^x\left|\begin{array}{ccc}-e^{-x}& -\sin x& \cos x\\ e^{-x}& -\cos x& -\sin x\\ -e^{-x}& \sin x& -\cos x\end{array}\right|-e^{-x}\left|\begin{array}{ccc}{e}^x& -\sin x& \cos x\\ e^x& -\cos x& -\sin x\\ e^x& \sin x& -\cos x\end{array}\right|+\cos x\left|\begin{array}{ccc}{e}^x& -e^{-x}& \cos x\\ e^x& e^{-x}& -\sin x\\ e^x& -e^{-x}& -\cos x\end{array}\right|-\sin x\left|\begin{array}{ccc}{e}^x& -e^{-x}& -\sin x\\ e^x& e^{-x}& -\cos x\\ e^x& -e^{-x}& \sin x\end{array}\right| \\ \mathbf{=}{\mathbf{e}}^{\mathbf{x}}\left(\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\right)\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\left(\mathbf{2}{\mathbf{e}}^{\mathbf{x}}\right)\mathbf{+}\mathbf{cosx}\left(\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{cosx}\right)\mathbf{-}\mathbf{sinx}\left(\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{sinx}\right) \\ \mathbf{=}\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}{\mathbf{cos}}^{\mathbf{2}}\mathbf{x}\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}{\mathbf{sin}}^{\mathbf{2}}\mathbf{x} \\ \mathbf{=}\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{-}\mathbf{4}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\left({\mathbf{cos}}^{\mathbf{2}}\mathbf{x}\mathbf{+}{\mathbf{sin}}^{\mathbf{2}}\mathbf{x}\right) \\ \mathbf{=}\mathbf{-}\mathbf{8}{\mathbf{e}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}} \end{gathered}$$

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 $\mathbf{y}\mathbf{=}{\mathbf{Ae}}^{\mathbf{x}}\mathbf{+}{\mathbf{Be}}^{\mathbf{-}\mathbf{x}}\mathbf{+}\mathbf{Ccosx}\mathbf{+}\mathbf{Dsinx}.$