Euler's method for coupled systems Problem set A

The variables \(x\) and \(y\) satisfy \(\dfrac{dx}{dt}=x-y+t\) and \(\dfrac{dy}{dt}=2x-t\), with \(x(0)=1\) and \(y(0)=2\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

The variables \(x\) and \(y\) satisfy \(\dfrac{dx}{dt}=xy\) and \(\dfrac{dy}{dt}=x-y^2\), with \(x(0)=1\) and \(y(0)=1\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

The variables \(x\) and \(y\) satisfy \(\dfrac{dx}{dt}=y\cos t\) and \(\dfrac{dy}{dt}=-x\sin t\), with \(x(0)=2\) and \(y(0)=1\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

Two species of plankton compete in a pond. Their numbers \(x\) and \(y\) (in thousands) at time \(t\) days satisfy \(\dfrac{dx}{dt}=0.8x-0.1x^2-0.3xy\) and \(\dfrac{dy}{dt}=0.6y-0.1y^2-0.1xy\), with \(x(0)=2\) and \(y(0)=3\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

Two tanks hold salt solution. The mass of salt \(x\) kg in tank 1 and \(y\) kg in tank 2 after \(t\) minutes satisfy \(\dfrac{dx}{dt}=2+0.5t-0.4x+0.1y\) and \(\dfrac{dy}{dt}=0.4x-0.6y\), with \(x(0)=1\) and \(y(0)=2\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

Two lakes receive pollution. The amounts \(x\) and \(y\) (in tonnes) in the lakes at time \(t\) months satisfy \(\dfrac{dx}{dt}=3\sin t-0.5x+0.3y\) and \(\dfrac{dy}{dt}=0.5x-0.4y\), with \(x(0)=0\) and \(y(0)=1\). Work in radians. Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

The populations of two species of beetle, \(x\) and \(y\) (in thousands), satisfy \(\dfrac{dx}{dt}=x(1-y)\) and \(\dfrac{dy}{dt}=y(x-1)\), with \(x(0)=1.5\) and \(y(0)=0.5\), where \(t\) is in years. Use Euler's method with \(h=0.25\) to estimate \(x\) and \(y\) when \(t=1\).

1. (a) Find the estimates of \(x(1)\) and \(y(1)\).
2. (b) Which of the two is larger at \(t=1\)?

The system \(\dfrac{dx}{dt}=ax+y+t\), \(\dfrac{dy}{dt}=x-by\) has \(x(0)=2\) and \(y(0)=1\), where \(a\) and \(b\) are constants. Euler's method with \(h=0.1\) gives \(x_1=2.35\) and \(y_1=1.12\).

1. (a) Find \(a\) and \(b\).
2. (b) Find \(x_2\).