Givet en differentialekvation,
d
y
d
t
=
f
(
t
,
y
)
,
y
(
t
0
)
=
y
0
,
{\displaystyle {\frac {dy}{dt}}=f(t,y),\quad y(t_{0})=y_{0},}
så uttrycks trapetsmetoden som
y
i
+
1
=
y
i
+
h
f
(
t
i
,
y
i
)
+
f
(
t
i
+
1
,
y
i
+
1
)
2
,
t
i
+
1
=
t
i
+
h
,
{\displaystyle y_{i+1}=y_{i}+h{\frac {f(t_{i},y_{i})+f(t_{i+1},y_{i+1})}{2}},\quad {t_{i+1}=t_{i}+h},}
Där
y
n
≈
y
(
t
n
)
{\displaystyle y_{n}\approx y(t_{n})}
n
>
0
{\displaystyle n>0}
y
i
+
1
{\displaystyle y_{i+1}}
Newtons metod , eller Eulers metod,om
f
(
t
,
y
)
{\displaystyle f(t,y)}
icke-linjär i
y
{\displaystyle y}
Om vi integrerar differentialekvationen från
t
n
{\displaystyle t_{n}}
t
n
+
1
{\displaystyle t_{n+1}}
insättningsformeln )
y
(
t
n
+
1
)
−
y
(
t
n
)
=
∫
t
n
t
n
+
1
f
(
t
,
y
(
t
)
)
d
t
.
{\displaystyle y(t_{n+1})-y(t_{n})=\int _{t_{n}}^{t_{n+1}}f(t,y(t))\,\mathrm {d} t.}
Denna integral kan approximeras med trapetsregeln ,
∫
t
n
t
n
+
1
f
(
t
,
y
(
t
)
)
d
t
≈
h
f
(
t
n
,
y
(
t
n
)
)
+
f
(
t
n
+
1
,
y
(
t
n
+
1
)
)
2
,
{\displaystyle \int _{t_{n}}^{t_{n+1}}f(t,y(t))\,\mathrm {d} t\approx h{\frac {f(t_{n},y(t_{n}))+f(t_{n+1},y(t_{n+1}))}{2}},}
vilket tillsammans med
y
n
≈
y
(
t
n
)
{\displaystyle y_{n}\approx y(t_{n})}
y
n
+
1
≈
y
(
t
n
+
1
)
{\displaystyle y_{n+1}\approx y(t_{n+1})}
y
i
+
1
=
y
i
+
h
f
(
t
i
,
y
i
)
+
f
(
t
i
+
1
,
y
i
+
1
)
2
,
t
i
+
1
=
t
i
+
h
,
{\displaystyle y_{i+1}=y_{i}+h{\frac {f(t_{i},y_{i})+f(t_{i+1},y_{i+1})}{2}},\quad {t_{i+1}=t_{i}+h},}
