| Exam Board | SPS |
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| Module | SPS FM Pure (SPS FM Pure) |
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| Year | 2023 |
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| Session | June |
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| Topic | Differential equations |
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13. (i) Solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + y ) ( 1 - x ) ,$$
given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\), where f is a function to be determined.
(ii) By considering the sign of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) near \(( 1,1 )\), or otherwise, show that this point is a maximum point on the curve \(y = \mathrm { f } ( x )\).
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