- The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } - x - 1$$
where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and \(y = 0\) at \(x = 0\)
Use the approximations
$$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$
with \(h = 0.1\) to find an estimate for the value of \(y\) at \(x = 0.2\)