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