FP2 June 2013 Q7

Notes

M1 for attempting to differentiate \(y = xv\) to get \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) – product rule must be used

M1 for differentiating their \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) to obtain an expression for \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) – product rule must be used

A1 for \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = \dfrac{\mathrm{d}v}{\mathrm{d}x} + \dfrac{\mathrm{d}v}{\mathrm{d}x} + x\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2}\)

M1 for substituting their \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) and \(y = xv\) in the original equation to obtain a differential equation in \(v\) and \(x\)

M1 for collecting the terms to have at most a 4 term equation – 4 terms only if a previous error causes \(\dfrac{\mathrm{d}v}{\mathrm{d}x}\) to be included, otherwise 3 terms

A1cao and cso for \(4\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 4v = x\) *

Alternative for (a)

M1 for writing \(v = \dfrac{y}{x}\) and attempting to differentiate by quotient or product rule to get \(\dfrac{\mathrm{d}v}{\mathrm{d}x}\)

M1 for differentiating their \(\dfrac{\mathrm{d}v}{\mathrm{d}x}\) to obtain an expression for \(\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2}\) – product or quotient rule must be used

A1 for \(\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} = \dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \dfrac{1}{x} - \dfrac{\mathrm{d}y}{\mathrm{d}x} \times \dfrac{1}{x^2} - \dfrac{\mathrm{d}y}{\mathrm{d}x} \times \dfrac{1}{x^2} + 2y \times \dfrac{1}{x^3}\)

M1 for multiplying their \(\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2}\) by \(x^3\)

M1 for multiplying by 4 and adding \(4x^2y\) to each side and equating to \(x^4\) (as rhs is now identical to the original equation.

A1cao and cso for \(4\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 4v = x\) *

Notes

M1 for forming the auxiliary equation and attempting to solve

A1 for \(\lambda^2 = -1\) oe

A1 for the complementary function in either form. Award for a correct CF even if \(\lambda = \mathrm{i}\) only is shown.

M1 for trying one of \(v = kx,\ k \neq 1\) or \(v = kx + l\) and \(v = mx^2 + kx + l\) as a PI and obtaining \(\dfrac{\mathrm{d}v}{\mathrm{d}x}\) and \(\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2}\)

M1dep for substituting their differentials in the equation \(4\dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} + 4v = x\). Award M0 if the original equation is used. Dep on 2nd M mark of (b)

A1cao for obtaining the correct result (either form)

Notes

B1ft for reversing the substitution to get \(y = x\left(C\cos x + D\sin x + \dfrac{1}{4}x\right)\) \(\left(\text{or } y = x\left(A\mathrm{e}^{\mathrm{i}x} + B\mathrm{e}^{-\mathrm{i}x} + \dfrac{1}{4}x\right)\right)\) follow through their answer to (b)