91Ó°ÊÓ

To obtain the characteristic equation of the given differential equation, we replace each term, i.e.,\(y^{(n)}\), with \(r^n\) in the equation. For the given differential equation, the characteristic equation is: \[r^4+1=0\]

To proceed, we will find the roots of the characteristic equation. The characteristic equation is a quartic equation, which can be rewritten as \(r^4=-1\). Since we are looking for the 4 root, we will separate it into its real and imaginary parts using Euler's formula. Euler's formula states that \(e^{i\theta} = \cos{\theta} + i\sin{\theta}\), therefore \(r = e^{i\theta} = \pm \sqrt[4]{1}\cdot e^{i(\theta + 2k\pi)}\) Since we need 4 different roots, k can have values 0, 1, 2, and 3. Thus, our roots will be: \[r_1=\sqrt[4]{1}\cdot e^{i(-\frac{\pi}{4}+\frac{0\pi}{2})} = e^{i\frac{-\pi}{4}}\] \[r_2=\sqrt[4]{1}\cdot e^{i(-\frac{\pi}{4}+\frac{2\pi}{2})} = e^{i\frac{3\pi}{4}}\] \[r_3=\sqrt[4]{1}\cdot e^{i(-\frac{\pi}{4}+\frac{4\pi}{2})} = e^{i\frac{5\pi}{4}}\] \[r_4=\sqrt[4]{1}\cdot e^{i(-\frac{\pi}{4}+\frac{6\pi}{2})} = e^{i\frac{7\pi}{4}}\]

The general solution of a linear homogeneous differential equation with constant coefficients is given by the following formula: \[y(x) = \sum_{j=1}^{n} C_j e^{r_jx}\cdot e^{i(\theta_j + 2kj\pi)}\] where \(C_j\) are constants, \(r_j\) are the roots of the characteristic equation, and \(\theta_j>0\). Using the roots we found in Step 2, the general solution of the given differential equation is: \[y(x) = C_1 e^{ix(-\frac{\pi}{4})} + C_2 e^{ix(\frac{3\pi}{4})} + C_3 e^{ix(\frac{5\pi}{4})} + C_4 e^{ix(\frac{7\pi}{4})}\] To obtain real-valued functions, we can rewrite the general solution using Euler's formula as: \[y(x) = C_1 \left(\cos{(-\frac{\pi}{4}x)}+i\sin{(-\frac{\pi}{4}x)}\right) + C_2 \left(\cos{(\frac{3\pi}{4}x)}+i\sin{(\frac{3\pi}{4}x)}\right) + C_3 \left(\cos{(\frac{5\pi}{4}x)}+i\sin{(\frac{5\pi}{4}x)}\right) + C_4 \left(\cos{(\frac{7\pi}{4}x)}+i\sin{(\frac{7\pi}{4}x)}\right)\] Finally, by separating real and imaginary parts and setting only the real part as the solution, we have: \[y(x) = A_1\cos{(-\frac{\pi}{4}x)} + A_2 \sin{(-\frac{\pi}{4}x)} + A_3 \cos{(\frac{3\pi}{4}x)} + A_4\sin{(\frac{3\pi}{4}x)}+ B_1 \cos{(\frac{5\pi}{4}x)} + B_2\sin{(\frac{5\pi}{4}x)} + B_3\cos{(\frac{7\pi}{4}x)} + B_4\sin{(\frac{7\pi}{4}x)}\] Here, \(A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4\) are constants that will depend on the initial or boundary conditions given for the specific problem.