| Exam Board | Edexcel |
|---|
| Module | FP2 (Further Pure Mathematics 2) |
|---|
| Topic | First order differential equations (integrating factor) |
|---|
14. (a) Use the substitution \(y = v x\) to transform the equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + y ) ( x + y ) } { x ^ { 2 } } , x > 0$$
into the equation
$$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( 2 + v ) ^ { 2 }$$
(b) Solve the differential equation II to find \(v\) as a function of \(x\).
(c) Hence show that
$$y = - 2 x - \frac { x } { \ln x + c } , \text { where } c \text { is an arbitrary constant, }$$
is a general solution of the differential equation I.
[0pt]
[P4 January 2003 Qn 5]