Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate to get y dy/(1+y²) = dx, integrate both sides using a standard integral (½ln(1+y²)), apply initial condition, and rearrange for y². The integration is bookwork and the algebraic manipulation is routine, making this slightly easier than average.
4 Given that \(y = 2\) when \(x = 0\), solve the differential equation
$$y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + y ^ { 2 }$$
obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
Separate variables correctly and attempt to integrate one side
M1
Obtain terms \(\frac{1}{3}\ln(1 + y^2)\) and \(x\), or equivalent
A1 + A1
Evaluate a constant or use limits \(x = 0, y = 2\) with a solution containing terms \(\ln(1 + y^2)\) and \(x\), or equivalent
M1
Obtain any correct form of solution, e.g. \(\frac{1}{3}\ln(1 + y^2) = x + \frac{1}{3}\ln 5\)
A1
Rearrange and obtain \(y^2 = 5e^{2x} - 1\), or equivalent
A1
6
Separate variables correctly and attempt to integrate one side | M1 |
Obtain terms $\frac{1}{3}\ln(1 + y^2)$ and $x$, or equivalent | A1 + A1 |
Evaluate a constant or use limits $x = 0, y = 2$ with a solution containing terms $\ln(1 + y^2)$ and $x$, or equivalent | M1 |
Obtain any correct form of solution, e.g. $\frac{1}{3}\ln(1 + y^2) = x + \frac{1}{3}\ln 5$ | A1 |
Rearrange and obtain $y^2 = 5e^{2x} - 1$, or equivalent | A1 |
| | 6 |
4 Given that $y = 2$ when $x = 0$, solve the differential equation
$$y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + y ^ { 2 }$$
obtaining an expression for $y ^ { 2 }$ in terms of $x$.
\hfill \mbox{\textit{CAIE P3 2006 Q4 [6]}}