1995 Paper 1 Q8

Year: 1995
Paper: 1
Question Number: 8

Course: UFM Pure
Section: Second order differential equations

Difficulty: 1500.0 Banger: 1532.8

second order differential equation variable coefficients particular integral complementary function linear combination simultaneous equations superposition principle

Problem

Find functions \(\mathrm{f,g}\) and \(\mathrm{h}\) such that \[ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+\mathrm{f}(x)\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{g}(x)y=\mathrm{h}(x)\tag{\ensuremath{*}} \] is satisfied by all three of the solutions \(y=x,y=1\) and \(y=x^{-1}\) for \(0 < x < 1.\) If \(\mathrm{f,g}\) and \(\mathrm{h}\) are the functions you have found in the first paragraph, what condition must the real numbers \(a,b\) and \(c\) satisfy in order that \[ y=ax+b+\frac{c}{x} \] should be a solution of \((*)\)?

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

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Banger Rating: 1532.8

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Problem source

Find functions $\mathrm{f,g}$ and $\mathrm{h}$ such that 
\[
\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+\mathrm{f}(x)\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{g}(x)y=\mathrm{h}(x)\tag{\ensuremath{*}}
\]
is satisfied by all three of the solutions $y=x,y=1$ and $y=x^{-1}$
for $0 < x < 1.$ 
 
If $\mathrm{f,g}$ and $\mathrm{h}$ are the functions you have found
in the first paragraph, what condition must the real numbers $a,b$
and $c$ satisfy in order that 
\[
y=ax+b+\frac{c}{x}
\]
should be a solution of $(*)$?

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