Standard +0.8 This is a multi-part Further Maths question requiring homogeneous DE substitution, separable variables, and algebraic manipulation to reach a specific form. While the techniques are standard for FP2, the question demands careful execution across three connected parts with non-trivial algebra, placing it moderately above average difficulty.
12. (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 \(\boldsymbol { v }\) as a function of \(\boldsymbol { x }\)
(c) Hence show that \(\quad y = - 2 x - \frac { x } { \ln x + c }\), where \(c\) is an arbitrary constant, is a general solution of the differential equation I.
12. (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 $\boldsymbol { v }$ as a function of $\boldsymbol { x }$\\
(c) Hence show that $\quad y = - 2 x - \frac { x } { \ln x + c }$, where $c$ is an arbitrary constant, is a general solution of the differential equation I.\\
\hfill \mbox{\textit{Edexcel FP2 2003 Q12 [10]}}