The differential equation EId4ydx4-Pd2ydx2-ky-0 is where E- I- P- K are functions of x -only

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Question

The differential equation $E I \frac{d^{4} y}{d x^{4}} + P \frac{d^{2} y}{d x^{2}} + k y = 0$ is

(where E, I, P, K are functions of x -only)

Linear of Fourth order

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Non-linear of Fourth order

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Non-Homogeneous

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Linear and Fourth degree

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Solution

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(\frac{d y}{d x})^{4} + 3 x \frac{d^{2} y}{d x^{2}} = 0

\frac{d^{4} y}{d x^{4}} + 3 \frac{d^{2} y}{d x^{2}} = 108 x^{2}

\frac{d^{2} y}{d x^{2}} = \sqrt[3]{1 + (\frac{d y}{d x})^{4}}

\frac{d^{2} y}{d x^{2}} = \sqrt[3]{1 - {(\frac{d y}{d x})}^{4}}

y = c_{1} e^{3 x} + c_{2} e^{2 x} + c_{3} e^{x}

c_{1}, c_{2}, c_{3}

\Rightarrow \frac{d^{3} y}{d x^{3}} - m \frac{d^{2} y}{d x^{2}} + n \frac{d y}{d x} - k y = 0

k + m - n

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