Give the solution of the following Differential Equation

Question

(y^(5) - 18 y''' + 65y')y = 0

Please show the process I know the answer is: y = C1 + C2 e^√5 x + C3 e^-√5 x + C4 e√13 x + C5 e^-√13 x

Answer 1

(y⁽⁵⁾ - 18y^''' + 65y')y = 0 y = 0 or y⁽⁵⁾ - 18y^''' + 65y^' = 0

Solve the auxiliary equation m⁵ - 18m³ + 65m = 0

m(m⁴ - 18m² + 65) = 0

m(m² - 13)(m² - 5) = 0

m(m - √13)(m + √13)(m - √5)(m + √5) = 0

m = 0, √13, -√13, √5, -√5

Does that help?