No links please Q y = xn-1 log x, Prove that x2 d2ydx2 + (3 - Maths - Differential Equations

Question

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Q     y   =   x n - 1   log   x
,   P r o v e   t h a t   x 2   d 2 y d x 2
  +   ( 3   -   2 n )   x d y d x   +
  ( n - 1 ) 2   y   =   0

Neha Sethi · Accepted Answer

Dear student
We have,y=xn-1logxDifferentiating w r t x
,we getdydx=n-1xn-2logx+xn-1×1x⇒dydx=n-1xn-2logx+xn-2⇒dydx=xn-2(n-1)logx+1Differentiating w
r t x
,we getd2ydx2=(n-2)xn-3(n-1)logx+1+xn-21xn-1⇒d2ydx2=(n-2)xn-3(n-1)logx+1+xn-3n-1⇒d2ydx2=xn-3(n-1)(n-2)logx+(n-2)+(n-1)⇒d2ydx2=xn-3(n-1)(n-2)logx+2n-3Consider,x2d2ydx2+3-2nxdydx+(n-1)2y=x2×xn-3(n-1)(n-2)logx+2n-3+(3-2n)x×xn-2(n-1)logx+1+(n-1)2xn-1logx=xn-1(n-1)(n-2)logx+2n-3+xn-13-2n(n-1)logx+1+xn-1n-12logx=xn-1n2-3n+2logx+2n-3+-2n2+5n-3logx+3-2n+n2+1-2nlogx=xn-1n2logx-3nlogx+2logx+2n-3-2n2logx+5nlogx-3logx+3-2n+n2logx+logx-2nlogx=0Hence,x2d2ydx2+3-2nxdydx+(n-1)2y=0
Regards

No links please Q . y = x n - 1 log x , P r o v e t h a t x 2 d 2 y d x 2 + ( 3 - 2 n ) . x d y d x + ( n - 1 ) 2 y = 0

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$Q . y = x^{n - 1} \log x, P r o v e t h a t x^{2} \frac{d^{2} y}{d x^{2}} + (3 - 2 n) . x \frac{d y}{d x} + (n - 1)^{2} y = 0$