The equation should be
x ( t + Δ t ) ≈ x ( t ) + Δ t 2 [ ( − x 3 ( t ) + sin ( t x ( t ) ) ) + ( − ( x ( t ) + f ( x ( t ) , t ) Δ t ) 3 + sin ( ( t + Δ t ) ( x ( t ) + f ( x ( t ) , t ) Δ t ) ) ) ] {\displaystyle x(t+\Delta t)\approx x(t)+{\frac {\Delta t}{2}}\left[\left(-x^{3}(t)+\sin \left(tx(t)\right)\right)+\left(-\left(x(t)+f(x(t),t)\Delta t\right)^{3}+\sin \left((t+\Delta t)(x(t)+f(x(t),t)\Delta t)\right)\right)\right]}
= x ( t ) + Δ t 2 [ ( − x 3 ( t ) + sin ( t x ( t ) ) ) + ( − ( x ( t ) + ( − x 3 ( t ) + sin ( t x ( t ) ) ) Δ t ) 3 + sin ( ( t + Δ t ) ( x ( t ) + ( − x 3 ( t ) + sin ( t x ( t ) ) ) Δ t ) ) ) ] {\displaystyle \qquad =x(t)+{\frac {\Delta t}{2}}\left[\left(-x^{3}(t)+\sin \left(tx(t)\right)\right)+\left(-\left(x(t)+(-x^{3}(t)+\sin(tx(t)))\Delta t\right)^{3}+\sin \left((t+\Delta t)(x(t)+(-x^{3}(t)+\sin(tx(t)))\Delta t)\right)\right)\right]}
Return to errata.
![{\displaystyle x(t+\Delta t)\approx x(t)+{\frac {\Delta t}{2}}\left[\left(-x^{3}(t)+\sin \left(tx(t)\right)\right)+\left(-\left(x(t)+f(x(t),t)\Delta t\right)^{3}+\sin \left((t+\Delta t)(x(t)+f(x(t),t)\Delta t)\right)\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c42334e2cec3c0850d126982f60b034093006c)
![{\displaystyle \qquad =x(t)+{\frac {\Delta t}{2}}\left[\left(-x^{3}(t)+\sin \left(tx(t)\right)\right)+\left(-\left(x(t)+(-x^{3}(t)+\sin(tx(t)))\Delta t\right)^{3}+\sin \left((t+\Delta t)(x(t)+(-x^{3}(t)+\sin(tx(t)))\Delta t)\right)\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d4f983a77525ed064d499f82c0c7495d158ae7)
