- The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 15 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 2 x$$
where \(y = 1\) at \(x = 0\) and where \(y = 2\) at \(x = 0.1\)
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 \(y\) when \(x = 0.3\)