To approximate y ( 1 / 4 ) {\displaystyle y(1/4)} , we use the Euler’s method y n + 1 = y n + Δ t f ( y n ) . {\displaystyle y_{n+1}=y_{n}+\Delta tf(y_{n}).} with a single step Δ t = 1 / 4 {\displaystyle \Delta t=1/4} and n = 0 {\displaystyle n=0} , where d y d t = f ( y ) {\displaystyle {\frac {dy}{dt}}=f(y)} . As y 0 = y ( 0 ) = 2 {\displaystyle y_{0}=y(0)=2} , we have
y ( 1 / 4 ) = y 1 = y 0 + Δ t ⋅ 2 ( 1 − y 0 ) = 2 + 1 4 ⋅ 2 ( 1 − 2 ) = 3 2 . {\displaystyle {\begin{aligned}y(1/4)=y_{1}&=y_{0}+\Delta t\cdot 2(1-y_{0})\\&=2+{\frac {1}{4}}\cdot 2(1-2)\\&={\frac {3}{2}}.\end{aligned}}}
