6 A differential equation is given by
$$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
- Show that the substitution \(x = \mathrm { e } ^ { 2 t }\) transforms this differential equation into
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 4 t ^ { 2 } + 5 \mathrm { e } ^ { - t }$$
- Hence find the general solution of the differential equation
$$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
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