A differential equation is given by
$$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$
Show that the substitution
$$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + x$$
transforms this differential equation into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$
(4 marks)
Find the general solution of
$$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$
giving your answer in the form \(u = \mathrm { f } ( x )\).
Hence find the general solution of the differential equation
$$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$
giving your answer in the form \(y = \mathrm { g } ( x )\).