Differential equation given

Questions that provide a differential equation relating derivatives (e.g., (x²+1)y'' = 2y² + (1-2x)y') and ask to derive higher-order relationships by differentiating the equation itself, then find the series.

2 questions

Edexcel F2 2018 June Q3
3. $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = 1$$
  1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \frac { 1 } { 2 } \left( a \frac { \mathrm {~d} y } { \mathrm {~d} x } + b x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + c \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } \right)$$ where \(a , b\) and \(c\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 2\)
  2. find a series solution for \(y\) in ascending powers of ( \(x - 2\) ), up to and including the term in \(( x - 2 ) ^ { 4 }\). Write each term in its simplest form.
  3. Use the solution to part (b) to find an approximate value for \(y\) when \(x = 2.1\), giving your answer to 3 decimal places.
Edexcel FP2 2008 June Q9
9. $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } + ( 1 - 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
  1. By differentiating equation (I) with respect to \(x\), show that $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = ( 1 - 4 x ) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 4 y - 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\),
  2. find the series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x _ { 3 }\).(4)
  3. Use your series to estimate the value of \(y\) at \(x = - 0.5\), giving your answer to two decimal places.(1)