Euler's method

Problem 6.299 · medium

1. Each step follows the tangent line: yₙ₊₁ = yₙ + h·f(tₙ, yₙ), tₙ₊₁ = tₙ + h.
   Reviewed
2. \[ -1 + 1 \cdot \frac{1}{4} \left(-1\right) = - \frac{5}{4} \]
   y1 = y0 + h·f(t0, y0) with t0 = 0, slope -1.✓ Proved
3. \[ - \frac{5}{4} + 1 \cdot \frac{1}{4} \left(-3\right) \frac{1}{4} = - \frac{23}{16} \]
   y2 = y1 + h·f(t1, y1) with t1 = 1/4, slope -3/4.✓ Proved
4. \[ - \frac{23}{16} + 1 \cdot \frac{1}{4} \left(-7\right) \frac{1}{16} = - \frac{99}{64} \]
   y3 = y2 + h·f(t2, y2) with t2 = 1/2, slope -7/16.✓ Proved
5. \[ - \frac{99}{64} + 1 \cdot \frac{1}{4} \left(-3\right) \frac{1}{64} = - \frac{399}{256} \]
   y4 = y3 + h·f(t3, y3) with t3 = 3/4, slope -3/64.✓ Proved