State the order of the given ordinary differential equation (1-x)y'' - 8xy' + 2y = cos(x) Determine whether the equation is linear or non-linear by matching it with (6) in section (1.1)
State the order of the given ordinary differential equation t^7 y^5 - t^4 y'' + 5y = 0 Determine whether the equation is linear or non linear by matching it with (6) in section (1.1)
Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. dy/dt + 30y = 24 ; y = 4/5 - 4/5 e^(-30t)
Verify that the indicated function y = ?(x) is an explicit solution of the given first-order differential equation y' = 2xy^2, y = 1/(25-x2) When y= 1/(25-x2), y'= Thus, in terms of x Since the left and right hand sides of the differential equation are equal when 1/(25 -x2) is substituted for y, y = 1/(25 -x2) is a solution. Proceed as in Example 4 of Section 1.1; by considering ? simply as a function, give its domain. (Enter your answer using interval notation.)
In this problem y=1/(1+c1 e^-x) is a one parameter family of solutions of the first order differential equations y' = y-y^2. Find a solution of the first order IVP consisting of this differential equation and the given initial condition. y(0) = -1/6
y=1/(x^2+c) is a one parameter family of solutions of the first order differential equation y'+2xy^2=0. Find a solution of the first order IVP consisting of this differential equation and the given initial condition. y(-3)=1/7
In this problem, x = c1 cos(t) + c2 sin(t) is a two-parameter family of solutions of the second-order DE x'' + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. x(0) = -1 and x'(0) = 9.
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0,y0) in the region. dy/dx=y^(4/5) There is a unique solution in any rectangular region where |y|<1 There is a unique solution in any rectangular region where x>0 There is a unique solution in any rectangular region excluding the origin There is a unique solution in any rectangular region where y=/0 There is a unique solution in entire xy plane
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0,y0) in the region. dy/dx = sqrt(xy) There is a unique solution in the entire xy plane There is a unique solution in any rectangular region where x<0 and y>0 There is a unique solution in any rectangular region where x>0 and y>0 or x>0 and y<0 There is a unique solution in any rectangular region where x>0 and y>0 or x<0 and y<0 There is a unique solution in the region x≤y
Determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0,y0) in the region. (4-y^2)y' = x^2 (1) A unique solution exists in the regions y < -2,-2 < y < 2,and y > 2. (2) A unique solution exists in the entire xy-plane. (3) A unique solution exists in the region y < 2. (4) A unique solution exists in the region y >
Determine theorem 1.2.1 guarantees that the differential equation y' = sqrt(y^2-25) possesses a unique solution through the given point (1,8).
Determine theorem 1.2.1 guarantees that the differential equation y' = sqrt(y^2-9) possesses a unique solution through the given point (6,3).
Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0 (see Equation (1) of Section 1.3). Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0. (Use P for P(t).) dP dt What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0?
Suppose a student carrying a flu virus returns to an isolated college campus of 7000 students. Determine a differential equation governing the number of students x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between students with the flu and students who have not yet contracted it. (Use k>0, for the constant of proportionality and x for x(t))
Suppose that a large mixing tank initially holds 2500 L of water in which 70 kg of salt have been dissolved. Pure water is pumped into the tank at a rate of 25 L/min, and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0. What is A(0)?
Suppose that a large mixing tank initially holds 1400 L of water in which 25 kg of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 10 L/min, and when the solution is well stirred, it is then pumped out at a slower rate of 7.5 L/min. If the concentration of the solution entering is 0.45 kg/L, determine a differential equation for the amount of salt A(t) in the tank at time t > 0.