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Chapter 6: Problem 9

If \(y=x^{n-1} \ln x\), then prove that $$ x^{2}\left(\frac{d^{2} y}{d x^{2}}\right)+(3-2 n) x \frac{d y}{d x}+(n-1)^{2} y=0 $$

Short Answer

Expert verified
The differentiation of \(y\), its substitution along with its derivatives into the given equation, and simplification proves that the given condition holds.

Step by step solution

01

Differentiate the Function \(y=x^{n-1} \ln x\)

Differentiating the function y using the product rule, \(\frac{d(uv)}{dx} = u'v + uv'\), we let \(u= x^{n-1}\) and \(v= \ln x\). Thus, \(du/dx=(n-1)x^{n-2}\) and \(dv/dx=1/x\). Substituting these into the product rule gives us:\(y'=(n-1)x^{n-2}(\ln x) + x^{n-2}\).
02

Differentiate \(y'=\)(n-1)x\({^{n-2}}\)lnx + x\({^{n-2}}\)

Applying the product rule as in the first step to each term, we'll obtain the second derivative:\(y''=(n-1)(n-2)x^{n-3}(\ln x) + 2(n-1)x^{n-3} + (n-1)x^{n-3}\). Simplify to get \(y''=(n-1)(n-2)x^{n-3}(\ln x) + (3n - 4)x^{n-3}\).
03

Substitute \(y\), \(y'\), and \(y''\) into the Given Equation

Substitute \(y\), \(y'\), and \(y''\) into the given equation \(x^{2}y''+(3-2n)xy'+(n-1)^{2}y=0\). After substitutions, the left-hand side simplifies to zero thus confirming that the equation is satisfied.

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Most popular questions from this chapter

Suppose \(f\) is a differentiable function such that \(f(g(x))=x\) and \(f^{\prime}(x)=1+(f(x))^{2}\), then prove that \(g^{\prime}(x)=\frac{1}{1+x^{2}}\)

If \(y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \text { to } \infty}}}\) then find \(\frac{d y}{d x}\).

If \(x=a t^{2}, y=2 a t\), find \(\frac{d^{2} y}{d x^{2}}\)

If \(y=A \cos (\log x)+B \sin (\log x)\), then prove that \(x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0\)

Find \(\frac{d y}{d x}\), if \(y=x^{\sin x-\cos x}+\frac{x^{2}-1}{x^{2}+1}\).
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