Question

11:32 83 A2 paper 2 Further... Q8 Given that \( y \) is a function of \( x \) and that \( x=\mathrm{e}^{u} \), show that \[ x \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} u} \quad \text { and } \quad x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{dx} x^{2}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} u} \text {. } \] Given also that \[ x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+3 x \frac{\mathrm{d} y}{\mathrm{~d} x}+17 y=34 \ln x+21, \] deduce that \[ \frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}+2 \frac{\mathrm{d} y}{\mathrm{~d} u}+17 y=34 u+21 \] Find \( y \) in terms of \( x \) given that \( y=0 \) and \( \frac{d y}{d x}=-1 \) when \( x=1 \). [3+2+6] \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( 19 \mid \) Page \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( 2 / 3 \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \)

          11:32

83
A2 paper 2 Further...
Q8
Given that \( y \) is a function of \( x \) and that \( x=\mathrm{e}^{u} \), show that
\[
x \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} u} \quad \text { and } \quad x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{dx} x^{2}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} u} \text {. }
\]

Given also that
\[
x^{2} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+3 x \frac{\mathrm{d} y}{\mathrm{~d} x}+17 y=34 \ln x+21,
\]
deduce that
\[
\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}+2 \frac{\mathrm{d} y}{\mathrm{~d} u}+17 y=34 u+21
\]

Find \( y \) in terms of \( x \) given that \( y=0 \) and \( \frac{d y}{d x}=-1 \) when \( x=1 \). [3+2+6]
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\( 19 \mid \) Page
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11:32

83
A2 paper 2 Further...
Q8
Given that y is a function of x and that x=e^u, show that

x (d y)/( d x)=(d y)/( d u) and x^2(d^2 y)/(dx x^2)=(d^2 y)/( d u^2)-(d y)/( d u).

Given also that

x^2(d^2 y)/( d x^2)+3 x (d y)/( d x)+17 y=34 ln x+21,

deduce that

(d^2 y)/( d u^2)+2 (d y)/( d u)+17 y=34 u+21

Find y in terms of x given that y=0 and (d y)/(d x)=-1 when x=1. [3+2+6]

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