Further Mathematics 9231: Differential equations Topical Past Paper Questions

It is given that $ v = xy^{3} $ and $$ xy^{2}\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+2xy\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}+(3x+2)y^{2}\frac{\mathrm{d}y}{\mathrm{d}x}+(x+1)y^{3}=x+5. $$ (a) Show that $$ \frac{1}{3}\frac{\mathrm{d}^{2}v}{\mathrm{d}x^{2}}+\frac{\mathrm…

Find the particular solution of the differential equation $$ \frac{\mathrm{d}y}{\mathrm{d}x}+\frac{1+x}{1-2x-x^{2}}y=1 $$ given that $ y = \pi $ when x = 0. Give your answer in an exact form. [10]

(a) Show that an appropriate integrating factor for $$ \sqrt{x^{2}+16}\frac{\mathrm{d}y}{\mathrm{d}x}+y=x\sqrt{x^{2}+16} $$ $$ \mathrm{i s}\;\tfrac{1}{4}x+\tfrac{1}{4}\sqrt{x^{2}+16}. $$ (b) Hence find the solution of the differential equation $$ \sqrt{x^{2}+16}\frac{\mathrm{d}y}…