find four linearly independent solutions of the equation y^iv + wy=0 in cases w=0 ,w<0,w>0

We are tasked with finding four linearly independent solutions to the differential equation

\[
y^{(iv)} + wy = 0
\]

in three cases: \( w = 0 \), \( w < 0 \), and \( w > 0 \).

### General Approach:
The given equation is a **fourth-order linear homogeneous differential equation**. To solve it, we first convert it to its characteristic form:

\[
r^4 + w = 0
\]

This simplifies to:

\[
r^4 = -w
\]

Now, we solve this equation for each case of \( w \).

---

### 1. Case 1: \( w = 0 \)
When \( w = 0 \), the characteristic equation becomes:

\[
r^4 = 0
\]

The only solution is \( r = 0 \), with multiplicity 4. Therefore, the general solution consists of four linearly independent solutions of the form:

\[
y(x) = c_1 + c_2 x + c_3 x^2 + c_4 x^3
\]

Thus, the four linearly independent solutions are:

\[
1, \, x, \, x^2, \, x^3
\]

---

### 2. Case 2: \( w < 0 \)
Let \( w = -\lambda \) where \( \lambda > 0 \). The characteristic equation becomes:

\[
r^4 = \lambda
\]

This equation has four distinct roots, which are the fourth roots of \( \lambda \):

\[
r = \pm \sqrt[4]{\lambda}, \quad r = \pm i\sqrt[4]{\lambda}
\]

These roots are complex numbers, and thus, the general solution includes real and imaginary parts. The corresponding four linearly independent solutions are:

\[
\cosh\left(\sqrt[4]{\lambda} x\right), \, \sinh\left(\sqrt[4]{\lambda} x\right), \, \cos\left(\sqrt[4]{\lambda} x\right), \, \sin\left(\sqrt[4]{\lambda} x\right)
\]

---

### 3. Case 3: \( w > 0 \)
Let \( w = \lambda \) where \( \lambda > 0 \). The characteristic equation becomes:

\[
r^4 = -\lambda
\]

This equation has complex roots:

\[
r = \pm \sqrt[4]{\lambda} i, \quad r = \pm \sqrt[4]{\lambda}
\]

The general solution involves trigonometric and hyperbolic functions, yielding four linearly independent solutions:

\[
\cos\left(\sqrt[4]{\lambda} x\right), \, \sin\left(\sqrt[4]{\lambda} x\right), \, \cosh\left(\sqrt[4]{\lambda} x\right), \, \sinh\left(\sqrt[4]{\lambda} x\right)
\]

---

### Summary of Solutions:
1. **For \( w = 0 \)**: \( 1, x, x^2, x^3 \)
2. **For \( w < 0 \)**: \( \cosh(\sqrt[4]{\lambda} x), \sinh(\sqrt[4]{\lambda} x), \cos(\sqrt[4]{\lambda} x), \sin(\sqrt[4]{\lambda} x) \)
3. **For \( w > 0 \)**: \( \cos(\sqrt[4]{\lambda} x), \sin(\sqrt[4]{\lambda} x), \cosh(\sqrt[4]{\lambda} x), \sinh(\sqrt[4]{\lambda} x) \)

---

Would you like a deeper explanation of any part of this solution? Here are 5 related questions you might want to explore:

1. How do we compute the general solution for higher-order linear differential equations?
2. What is the significance of linear independence in the context of differential equations?
3. Can we explore the derivation of hyperbolic and trigonometric functions from complex roots?
4. How do we classify the nature of solutions based on the characteristic equation's roots?
5. What methods can we use to verify the linear independence of the solutions?

**Tip:** For higher-order differential equations, always solve the characteristic equation first to determine the form of the general solution.

find four linearly independent solutions of the equation y^iv + wy = 0 in cases w = 0, w < 0, w > 0

Explore the step-by-step process of finding four linearly independent solutions to the fourth-order differential equation y^(iv) + wy = 0 for different values of w: w = 0, w < 0, and w > 0. This guide covers solving characteristic equations and understanding the nature of the solutions in each case.

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