Euler's method for coupled systems Problem set B

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

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

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

A lake has prey fish \(x\) and predator fish \(y\) (in hundreds). At time \(t\) months, \(\dfrac{dx}{dt}=0.9x-0.3xy\) and \(\dfrac{dy}{dt}=0.25xy-0.5y\), with \(x(0)=4\) and \(y(0)=2\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

Two rooms are heated. Their temperatures above the outside, \(x\) and \(y\) in \(^\circ\text{C}\), at time \(t\) hours satisfy \(\dfrac{dx}{dt}=3+t-0.5x+0.3y\) and \(\dfrac{dy}{dt}=0.3x-0.4y\), with \(x(0)=4\) and \(y(0)=1\). Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

Two greenhouses are heated by a daily cycle. Their temperatures above the outside, \(x\) and \(y\) in \(^\circ\text{C}\), at time \(t\) (in suitable units) satisfy \(\dfrac{dx}{dt}=2\cos t-0.4x+0.1y\) and \(\dfrac{dy}{dt}=0.4x-0.3y\), with \(x(0)=1\) and \(y(0)=0\). Work in radians. Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) when \(t=0.5\).

The numbers of two kinds of bacteria, \(x\) and \(y\) (in millions), satisfy \(\dfrac{dx}{dt}=x(2-y)\) and \(\dfrac{dy}{dt}=y(x-2)\), with \(x(0)=3\) and \(y(0)=1\), where \(t\) is in hours. 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 \(\dfrac{dx}{dt}=ay-x+t\), \(\dfrac{dy}{dt}=bx+y\) has \(x(0)=1\) and \(y(0)=2\), where \(a\) and \(b\) are constants. Euler's method with \(h=0.1\) gives \(x_1=1.3\) and \(y_1=2.5\).

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