Euler's method Problem set B One question at a time

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=2y\) and \(y(0)=1.5\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(2y_n\right)\)
• \(y_1=1.5+0.2\times(3)=2.1\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 2.1), (0.4, 2.94), (0.6, 4.116), (0.8, 5.7624), (1, 8.0674)
• y(1) ≈ 8.07

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=y-3x\) and \(y(0)=2\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(y_n-3x_n\right)\)
• \(y_1=2+0.2\times(2)=2.4\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 2.4), (0.4, 2.76), (0.6, 3.072), (0.8, 3.3264), (1, 3.5117)
• y(1) ≈ 3.51

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=2x+y\) and \(y(0)=−1\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(2x_n+y_n\right)\)
• \(y_1=−1+0.2\times(−1)=−1.2\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, −1.2), (0.4, −1.36), (0.6, −1.472), (0.8, −1.5264), (1, −1.5117)
• y(1) ≈ −1.51

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=\dfrac{2x}{y}\) and \(y(0)=3\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(\dfrac{2x_n}{y_n}\right)\)
• \(y_1=3+0.2\times(0)=3\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 3), (0.4, 3.0267), (0.6, 3.0795), (0.8, 3.1575), (1, 3.2588)
• y(1) ≈ 3.26

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=x^2y\) and \(y(0)=2\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(x_n^2y_n\right)\)
• \(y_1=2+0.2\times(0)=2\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 2), (0.4, 2.016), (0.6, 2.0805), (0.8, 2.2303), (1, 2.5158)
• y(1) ≈ 2.52

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=ye^{-2x}\) and \(y(0)=2\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(y_ne^{-2x_n}\right)\)
• \(y_1=2+0.2\times(2)=2.4\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 2.4), (0.4, 2.7218), (0.6, 2.9663), (0.8, 3.145), (1, 3.272)
• y(1) ≈ 3.27

The mass \(m\) grams of a radioactive sample satisfies \(\dfrac{dm}{dt}=-0.5m\) with \(m(0)=6\), where \(t\) is the time in days. Use Euler's method with \(h=0.2\) to estimate \(m(1)\).

• \(t_{n+1}=t_n+0.2\), \(m_{n+1}=m_n+0.2\left(-0.5m_n\right)\)
• \(m_1=6+0.2\times(−3)=5.4\)
• From the calculator, (t, m) for n = 1 to 5: (0.2, 5.4), (0.4, 4.86), (0.6, 4.374), (0.8, 3.9366), (1, 3.5429)
• m(1) ≈ 3.54 g

The number \(R\) (in hundreds) of students who know a rumour satisfies \(\dfrac{dR}{dt}=0.4R\left(1-\dfrac{R}{5}\right)\) with \(R(0)=0.5\), where \(t\) is the time in hours. Use Euler's method with \(h=0.2\) to estimate \(R(1)\).

• \(t_{n+1}=t_n+0.2\), \(R_{n+1}=R_n+0.2\left(0.4R_n\left(1-\dfrac{R_n}{5}\right)\right)\)
• \(R_1=0.5+0.2\times(0.18)=0.536\)
• From the calculator, (t, R) for n = 1 to 5: (0.2, 0.536), (0.4, 0.5743), (0.6, 0.6149), (0.8, 0.6581), (1, 0.7038)
• R(1) ≈ 0.704 hundred

A differential equation is \(\dfrac{dy}{dx}=x-ky\), where \(k\) is a constant, and \(y(0)=5\). Euler's method with \(h=0.2\) gives \(y_1=4.2\).

1. (a) Find the value of \(k\).

   • y₁ = 5 + 0.2(0 − k × 5) = 5 − k
   • 5 − k = 4.2, so k = 0.8

2. (b) Find \(y_2\).

   • x₁ = 0.2, y₁ = 4.2
   • y₂ = 4.2 + 0.2(0.2 − 0.8 × 4.2) = 3.568