First order differential equations (integrating factor)
4
A differential equation is given by
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$
Show that the substitution
$$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$
transforms this differential equation into
$$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
By using an integrating factor, find the general solution of
$$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
giving your answer in the form \(u = \mathrm { f } ( x )\).
Hence find the general solution of the differential equation
$$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$
giving your answer in the form \(y = \mathrm { g } ( x )\).