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Chapter 6: Problem 2

If \(y=\frac{a x+b}{x^{2}+c}\), then show that $$ \left(2 x \frac{d y}{d x}+y\right) \frac{d^{3} y}{d x^{3}}=3\left(x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}} $$

Short Answer

Expert verified
All three derivatives of the function \(\frac{a x+b}{x^{2}+c}\) with respect to \(x\), were calculated using the quotient rule of differentiation. The derivatives were then substituted back into the given differential equation. Upon simplifying both sides of the equation, it was confirmed that the two sides were indeed equal thus proving the given equation.

Step by step solution

01

Calculate the first derivative

The first derivative of \(y\) with respect to \(x\) is obtained by differentiating the function \(y=\frac{a x+b}{x^{2}+c}\). To solve this, we can use the quotient rule of differentiation, which is given by \(\frac{d}{dx}(\frac{u}{v}) = \frac{vu'-uv'}{v^2}\), where \(u=ax+b\) and \(v=x^2+c\). Calculating this gives \(y' = \frac{(x^2+c)(a) - (ax+b)(2x)}{(x^2+c)^2}\).
02

Calculate the second derivative

The second derivative of \(y\) with respect to \(x\) is calculated by differentiating the output of the previous step \(y'\) again. Again we need to apply the quotient rule here. Let \(u=(a(x^2+c) - (2ax+b))\) and \(v=(x^2+c)^2\). Calculate \(y''=\frac{(x^2+c)^2(0) - (a(x^2+c) - (2ax+b))(4x)}{((x^2+c)^2)^2}\).
03

Calculate the third derivative

The third derivative of \(y\) with respect to \(x\) is calculated by differentiating the output of the previous step \(y''\). Again we will be needing to use the quotient rule of differentiation here. Let \(u=(a(0) - 4x(ax+b))\), and \(v=((x^2+c)^2)^2\). Hence, calculate \(y'''=\frac{(((x^2+c)^2)^2)(0) - (a(x^2+c) - 4x(ax+b))(8x)}{(((x^2+c)^2)^2)^2}\).
04

Substituting the derivatives back into the given equation

Substitute the values of \(y\), \(y'\), \(y''\), and \(y'''\) in the given equation \(\left(2 x \frac{d y}{d x}+y\right) \frac{d^{3} y}{d x^{3}}=3\left(x\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\right) \frac{d^{2} y}{d x^{2}}\). Doing this, verify that both sides of the equation equate to each other, hence validating the original statement.

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Most popular questions from this chapter

If \(y=\tan ^{-1} x\), then prove that (i) \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\) (ii) \(\left(1+x^{2}\right) y_{n+2}+2(n+2) x y_{n+1}+n(n+1) y_{n}\) \(=0 .\)

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