Further Pure Mathematics WFM01: DIFFERENTIAL EQUATIONS Topical Past Paper Questions

(a) Given that $u = xy$, where $u$ is a function of $x$ ($x > 0$), show that $$ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{1}{x}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}x^{2}}-2\frac{\mathrm{d}y}{\mathrm{d}x}\right) $$ (b) Hence show that the transformation $u = xy$ transforms t…

(a) Show that the substitution $ z = \frac{1}{y^{2}} $ transforms the differential equation $$ 2\frac{\mathrm{d}y}{\mathrm{d}x}+(\cot x)y+(\tan x\sec x)y^{3}=0\quad0 (5) Given that $ y^{2}=\frac{4\sqrt{3}}{3} $ when $ x=\frac{\pi}{6} $ (c) determine the exact values of y when $ x…

(a) Given that $t = \ln x$, where $x > 0$, show that $$ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\mathrm{e}^{-2t}\left(\frac{\mathrm{d}^{2}y}{\mathrm{d}t^{2}}-\frac{\mathrm{d}y}{\mathrm{d}t}\right) $$ (b) Hence show that the transformation $ t = \ln x $, where x > 0, transforms th…