1.
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \cos x$$
- Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(x , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
At \(x = 0 , y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\)
- Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) at \(x = 0\)
- Express \(y\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).