1994 Paper 3 Q4

Year: 1994
Paper: 3
Question Number: 4

Course: LFM Pure
Section: Differential equations

Difficulty: 1700.0 Banger: 1484.7

differential equation first order separation of variables curve sketching parabola initial conditions common tangent singular solution family of curves

Problem

Find the two solutions of the differential equation \[ \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=4y \] which pass through the point \((a,b^{2}),\) where \(b\neq0.\) Find two distinct points \((a_{1},1)\) and \((a_{2},1)\) such that one of the solutions through each of them also passes through the origin. Show that the graphs of these two solutions coincide and sketch their common graph, together with the other solutions through \((a_{1},1)\) and \((a_{2},1)\). Now sketch sufficient members of the family of solutions (for varying \(a\) and \(b\)) to indicate the general behaviour. Use your sketch to identify a common tangent, and comment briefly on its relevance to the differential equation.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

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Banger Rating: 1484.7

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Problem source

Find the two solutions of the differential equation 
\[
\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=4y
\]
which pass through the point $(a,b^{2}),$ where $b\neq0.$

Find two distinct points $(a_{1},1)$ and $(a_{2},1)$ such that one
of the solutions through each of them also passes through the origin.
Show that the graphs of these two solutions coincide and sketch their
common graph, together with the other solutions through $(a_{1},1)$
and $(a_{2},1)$. 

Now sketch sufficient members of the family of solutions (for varying
$a$ and $b$) to indicate the general behaviour. Use your sketch
to identify a common tangent, and comment briefly on its relevance
to the differential equation.

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