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

Use Euler's method with \(h=0.2\) to estimate \(y(1)\), given \(\dfrac{dy}{dx}=3y\) and \(y(0)=0.5\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(3y_n\right)\)
• \(y_1=0.5+0.2\times(1.5)=0.8\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 0.8), (0.4, 1.28), (0.6, 2.048), (0.8, 3.2768), (1, 5.2429)
• y(1) ≈ 5.24

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

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(y_n-2x_n\right)\)
• \(y_1=1+0.2\times(1)=1.2\)
• 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 \(\dfrac{dy}{dx}=x-3y\) and \(y(0)=2\).

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(x_n-3y_n\right)\)
• \(y_1=2+0.2\times(−6)=0.8\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 0.8), (0.4, 0.36), (0.6, 0.224), (0.8, 0.2096), (1, 0.2438)
• y(1) ≈ 0.244

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

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(\dfrac{3x_n}{y_n}\right)\)
• \(y_1=2+0.2\times(0)=2\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 2), (0.4, 2.06), (0.6, 2.1765), (0.8, 2.3419), (1, 2.5469)
• y(1) ≈ 2.55

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

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(x_ny_n^2\right)\)
• \(y_1=1+0.2\times(0)=1\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 1), (0.4, 1.04), (0.6, 1.1265), (0.8, 1.2788), (1, 1.5405)
• y(1) ≈ 1.54

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

• \(x_{n+1}=x_n+0.2\), \(y_{n+1}=y_n+0.2\left(y_ne^{-x_n}\right)\)
• \(y_1=1+0.2\times(1)=1.2\)
• From the calculator, (x, y) for n = 1 to 5: (0.2, 1.2), (0.4, 1.3965), (0.6, 1.5837), (0.8, 1.7575), (1, 1.9155)
• y(1) ≈ 1.92

The concentration \(C\) mg/L of a drug in the blood satisfies \(\dfrac{dC}{dt}=-0.35C\) with \(C(0)=8\), where \(t\) is the time in hours. Use Euler's method with \(h=0.2\) to estimate \(C(1)\).

• \(t_{n+1}=t_n+0.2\), \(C_{n+1}=C_n+0.2\left(-0.35C_n\right)\)
• \(C_1=8+0.2\times(−2.8)=7.44\)
• From the calculator, (t, C) for n = 1 to 5: (0.2, 7.44), (0.4, 6.9192), (0.6, 6.4349), (0.8, 5.9844), (1, 5.5655)
• C(1) ≈ 5.57 mg/L

The number \(N\) (in thousands) of insects on an island satisfies \(\dfrac{dN}{dt}=0.5N\left(1-\dfrac{N}{8}\right)\) with \(N(0)=1\), where \(t\) is the time in months. Use Euler's method with \(h=0.2\) to estimate \(N(1)\).

• \(t_{n+1}=t_n+0.2\), \(N_{n+1}=N_n+0.2\left(0.5N_n\left(1-\dfrac{N_n}{8}\right)\right)\)
• \(N_1=1+0.2\times(0.4375)=1.0875\)
• From the calculator, (t, N) for n = 1 to 5: (0.2, 1.0875), (0.4, 1.1815), (0.6, 1.2822), (0.8, 1.3898), (1, 1.5047)
• N(1) ≈ 1.50 thousand

A differential equation is \(\dfrac{dy}{dx}=ky+x\), where \(k\) is a constant, and \(y(0)=2\). Euler's method with \(h=0.2\) gives \(y_1=2.6\).

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

   • y₁ = 2 + 0.2(k × 2 + 0) = 2 + 0.4k
   • 2 + 0.4k = 2.6, so k = 1.5

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

   • x₁ = 0.2, y₁ = 2.6
   • y₂ = 2.6 + 0.2(1.5 × 2.6 + 0.2) = 3.42