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05:0329–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.
y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ
42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.
b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)
y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1
Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].
b. Write the first step of Euler’s method to compute u1.
27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.
x′(t) = 2x − xy, y′(t) = −y + xy
29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.
b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.
y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t
brOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x² + y² = a²
b. The family of trajectories orthogonal to 2x² + y² = a² satisfies the differential equation dy/dx = y/(2x). Why?
