Given y1(t) = t^2 is a solution to: t^2y'' - 4ty' + 6y = 0, t > 0 find another solution using the method of reduction of order.

Answers

Answer 1

For a second order differential equation t²y'' - 4ty' + 6y = 0, general solution is of the form y(t) = c₁t² + c₂t³, so another solution other than y₁(t) = t² is y₂(t) = t³.

Given solution to t²y'' - 4ty' + 6y = 0 is

y₁(t) = t²

Using the method of reduction of order, let us assume, y₂(t) = v(t)y₁(t) is a solution to t²y'' - 4ty' + 6y = 0 for suitable choice of v(t). So,

y₂ = vt²

y₂' = 2vt + t²v'

y₂'' = 2v + 2tv' + 2tv' + t²v''

y₂'' = 2v + 4tv' + t²v''

Substituting this

t²×( 2v + 4tv' + t²v'') - 4t×(2vt + t²v') + 6×(vt²) = 0

t⁴v'' = 0

v'' = 0

v' = c, c is a constant

v = ct

Therefore, y₂(t) = ct×t²

y₂ (t) = ct³

Therefore general solution will be y(t) = c₁t² + c₂t³

To know more on second order differential equation

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La pregunta está incompleta:

Observa los puntajes de las respuestas de una evaluación que tuvo de 100 preguntas. Si Alfredo contestó correctamente 57 preguntas, 16 incorrectas y el resto las dejó en blanco, ¿qué puntuación obtuvo en la evaluación?

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Answers

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