2000 Paper 2 Q8

Year: 2000
Paper: 2
Question Number: 8

Course: UFM Pure
Section: First order differential equations (integrating factor)

Difficulty: 1600.0 Banger: 1500.1
differential equation first order separable equation exponential function asymptotic behaviour initial value problem integration

Problem

  1. Let \(y\) be the solution of the differential equation \[ \frac{\d y}{\d x} + 4x\e^{-x^2} {(y+3)}^{\frac12} = 0 \qquad (x \ge 0), \] that satisfies the condition \(y=6\) when \(x=0\). Find \(y\) in terms of \(x\) and show that \(y\to1\) as \(x \to \infty\).
  2. Let \(y\) be any solution of the differential equation \[ \frac{\d y}{\d x} -x\e^{6 x^2} (y+3)^{1-k} = 0 \qquad (x \ge 0). \] %that satisfies the condition \(y=6\) %when \(x=0\). Find a value of \(k\) such that, as \(x \to \infty\), \(\e^{-3x^2}y\) tends to a finite non-zero limit, which you should determine.
\noindent [The approximations, valid for small \(\theta\), \(\sin\theta \approx \theta\) and \(\cos\theta \approx 1-{\textstyle\frac12}\,\theta^2\) may be assumed.]

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1500.1

Banger Comparisons: 2

Show LaTeX source
Problem source
\begin{questionparts}
\item
Let $y$ be the solution of the differential equation
\[
\frac{\d y}{\d x} +  4x\e^{-x^2} {(y+3)}^{\frac12} = 0 \qquad (x \ge 0),
\]
that satisfies the condition $y=6$
when $x=0$.
Find $y$ in terms of $x$ and show that  
$y\to1$ 
as $x \to \infty$.
\item
Let $y$ be any solution of the differential equation
\[
\frac{\d y}{\d x}  -x\e^{6 x^2} (y+3)^{1-k} = 0 \qquad (x \ge 0).
\]
%that satisfies the condition $y=6$
%when $x=0$.
Find a value of  $k$ such that,
as $x \to \infty$,
$\e^{-3x^2}y$ 
tends to a finite non-zero  limit, which you should determine.
\end{questionparts}
\noindent
[The approximations, valid for small $\theta$, $\sin\theta \approx \theta$
and $\cos\theta \approx 1-{\textstyle\frac12}\,\theta^2$ may be assumed.]
Back to List