Euler's method Problem set A

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

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

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

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

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

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

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 \(C(1)\).

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 \(N(1)\).

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