| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Euler-Cauchy equations via exponential substitution |
| Difficulty | Challenging +1.2 This is a standard Euler-Cauchy equation problem from FP2 with a prescribed substitution method. Part (a) is routine verification of given derivatives using chain rule, part (b) is solving a standard constant-coefficient second-order DE with particular integral, and part (c) is back-substitution. While requiring multiple techniques and careful algebra, this follows a well-established template that FP2 students practice extensively, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.10e Second order non-homogeneous: complementary + particular integral |
7. (a) Show that the substitution $x = e ^ { u }$ transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = - x ^ { - 2 } , \quad x > 0$$
into the equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 2 y = - \mathrm { e } ^ { - 2 u }$$
(b) Find the general solution of the differential equation (II).\\
(c) Hence obtain the general solution of the differential equation (I) giving your answer in the form $y = \mathrm { f } ( x )$\\
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\hfill \mbox{\textit{Edexcel FP2 2016 Q7 [14]}}