Solve-the-differential-equation-x-2-D-2-2-y-x-2-1-x-

Question Number 90307 by niroj last updated on 22/Apr/20

Solve the differential equation .

(x^{2} D^{2} - 2) y = x^{2} + \frac{1}{x} .

Answered by TANMAY PANACEA. last updated on 22/Apr/20

x^{2} \frac{d^{2} y}{{d x}^{2}} - 2 y = x^{2} + \frac{1}{x}

x = e^{t}

\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x} = \frac{1}{e^{t}} \frac{d y}{d t}

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d t} (\frac{1}{e^{t}} \times \frac{d y}{d t}) \times \frac{d t}{d x} = \frac{1}{e^{t}} [\frac{1}{e^{t}} \times \frac{d^{2} y}{{d t}^{2}} - \frac{1}{e^{t}} \times \frac{d y}{d t}]

{(e^{t})}^{2} \frac{d^{2} y}{{d x}^{2}} = \frac{d^{2} y}{{d t}^{2}} - \frac{d y}{d t}

x^{2} \frac{d^{2} y}{{d x}^{2}} = \frac{d^{2} y}{{d t}^{2}} - \frac{d y}{d t}

s o

\frac{d^{2} y}{{d t}^{2}} - \frac{d y}{d t} - 2 y = e^{2 t} + e^{- t}

l e t y = e^{m t}

(m^{2} - m - 2) e^{m t} = 0 f o r C . F

(m - 2) (m + 1) = 0

C . F = {A e}^{2 t} + {B e}^{- t} = A x^{2} + \frac{B}{x}

P . I

l e t \frac{d}{d t} = θ

y = \frac{e^{2 t} + e^{- t}}{(θ^{2} - θ - 2)} = \frac{x^{2} + \frac{1}{x}}{(θ - 2) (θ + 1)} = \frac{1}{3} \times \frac{(θ + 1) - (θ - 2)}{(θ - 2) (θ + 1)} \times (x^{2} + \frac{1}{x})

= \frac{1}{3} [\frac{1}{θ - 2} - \frac{1}{θ + 1}] (x^{2} + \frac{1}{x})

= \frac{1}{3} \times [x^{2} \int x^{- 2 - 1} (x^{2} + \frac{1}{x}) d x - x^{- 1} \int x^{1 - 1} (x^{2} + \frac{1}{x}) d x]

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

= \frac{1}{3} [x^{2} \times l n x + x^{2} \times \frac{1}{x^{3} \times (- 3)} - \frac{1}{x} \times \frac{x^{3}}{3} - \frac{1}{x} l n x]

= \frac{x^{2} l n x}{3} - \frac{1}{9} \times \frac{1}{x} - \frac{x^{2}}{9} - \frac{l n x}{3 x}

= (x^{2} - \frac{1}{x}) \times \frac{l n x}{3} - \frac{1}{9} (x^{2} + \frac{1}{x})

y = {A x}^{2} + \frac{B}{x} + \frac{x^{2} l n x}{3} - \frac{l n x}{3 x} - \frac{x^{2}}{9} - \frac{1}{9 x}

y = C_{1} x^{2} + \frac{C_{2}}{x} + \frac{x^{2} l n x}{3} - \frac{l n x}{3 x}

Commented by niroj last updated on 22/Apr/20

I apriciate your trying effort but

Answer should make sure

y = C_{1} x^{- 1} + C_{2} x^{2} + \frac{1}{3} x^{2} \log x - \frac{1}{3 x} \log x .

Commented by TANMAY PANACEA. last updated on 22/Apr/20

i h a v e c o r r e c t e d \dots u s i n g D a n i e l a n d m u r r y d i f f

c a l b o o k

Commented by niroj last updated on 23/Apr/20

now done dear .

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