Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables to get dy/y² = 4xe^(-2x)dx, integrate both sides (right side needs integration by parts), then apply initial condition. While integration by parts adds a step beyond the most basic separable equations, this remains a routine textbook exercise with no conceptual challenges or novel insights required.
7 The variables \(x\) and \(y\) satisfy the differential equation
$$\mathrm { e } ^ { 2 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 x y ^ { 2 }$$
and it is given that \(y = 1\) when \(x = 0\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
Complete integration and obtain \(-2xe^{-2x} - e^{-2x}\)
A1
Evaluate a constant or use \(x=0\) and \(y=1\) as limits in a solution containing terms of the form \(\frac{p}{y}\), \(qxe^{-2x}\), \(re^{-2x}\), or equivalent
M1
Obtain \(y = \frac{e^{2x}}{2x+1}\), or equivalent expression for \(y\)
A1
ISW
## Question 7:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables correctly | B1 | $\int \frac{1}{y^2} dy = \int 4xe^{-2x} dx$ |
| $\int \frac{1}{y^2} dy = -\frac{1}{y}$ | B1 | OE |
| Commence the other integration and reach $axe^{-2x} + b\int e^{-2x} dx$ | M1 | |
| Obtain $-2xe^{-2x} + 2\int e^{-2x} dx$ or $-\frac{1}{2}xe^{-2x} + \frac{1}{2}\int e^{-2x} dx$ | A1 | SOI (might have taken out factor of 4) |
| Complete integration and obtain $-2xe^{-2x} - e^{-2x}$ | A1 | |
| Evaluate a constant or use $x=0$ and $y=1$ as limits in a solution containing terms of the form $\frac{p}{y}$, $qxe^{-2x}$, $re^{-2x}$, or equivalent | M1 | |
| Obtain $y = \frac{e^{2x}}{2x+1}$, or equivalent expression for $y$ | A1 | ISW |
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7 The variables $x$ and $y$ satisfy the differential equation
$$\mathrm { e } ^ { 2 x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 4 x y ^ { 2 }$$
and it is given that $y = 1$ when $x = 0$.\\
Solve the differential equation, obtaining an expression for $y$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2021 Q7 [7]}}