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x-5-t-x-8-t-2-x-2-ln-t-t-2-solve-the-differential-eqn-




Question Number 184683 by alcohol last updated on 10/Jan/23
x′′ − (5/t) x′ + (8/t^2 ) x = ((2 ln t)/t^2 )  solve the differential eqn
x5tx+8t2x=2lntt2solvethedifferentialeqn
Answered by qaz last updated on 10/Jan/23
t^2 x′′−5tx′+8x=2lnt  t=e^z ,  x_p =(1/(D(D−1)−5D+8))2z=((1/(D−4))−(1/(D−2)))z  =e^(4z) (1/D)e^(−4z) z−e^(2z) (1/D)e^(−2z) z=e^(4z) ∫e^(−4z) zdz−e^(2z) ∫e^(−2z) zdz  =(1/4)z+(3/(16))  ⇒x=C_1 e^(4z) +C_2 e^(2z) +(1/4)z+(3/(16))=C_1 t^4 +C_2 t^2 +(1/4)lnt+(3/(16))
t2x5tx+8x=2lntt=ez,xp=1D(D1)5D+82z=(1D41D2)z=e4z1De4zze2z1De2zz=e4ze4zzdze2ze2zzdz=14z+316x=C1e4z+C2e2z+14z+316=C1t4+C2t2+14lnt+316

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