a differential equation is given along with the field or problem area in which it arises....

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a differential equation is given along with the field or problem area in which it arises. Classify each as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ordinary differential equation, indicate whether the equation is linear or nonlinear 1. 4. dx² 2 gửi dr (mechanical vibrations, electrical circuits, seismology) (Hermite's equation, quantum-mechanical harmonic oscillator) 6. ax dy 2x +2y=0 dx +45 + 9x = 2 cos 3r dr + = 0 (Laplace's equation, potential theory, electricity, heat, aerodynamics) dy y(2-3x) dx x(1-3y) (competition between two species, ecology) dx 5. = k(4x)(1-x), where k is a constant dt (chemical reaction rates) + C, where C is a constant (brachistochrone problem, calculus of variations) dy 7. VI-y +2x- <=0 dx (Kidder's equation, flow of gases through a porous medium) 8. 9. 8 = x(1-x) dx dp = kp(P-p), where kand P are constants dr (logistic curve, epidemiology, economics) 11. 12. (deflection of beams) d'y 10. x + + xy =0 dx² dx (aerodynamics, stress analysis) a²N 1 aN + ar² rar aN ar (nuclear fission) + kN, where k is a constant dy -0.1(1-²) +9y=0 (van der Pol's equation, triode vacuum tube) In Problems 13-16, write a differential equation that fits the physical description. 13. The rate of change of the population p of bacteria at timer is proportional to the population at time 1. 14. The velocity at timer of a particle moving along a straight line is proportional to the fourth power of its position 1. 15. The rate of change in the temperature 7 of coffee at timer is proportional to the difference between the temperature M of the air at times and the tempera- ture of the coffee at time r. 16. The rate of change of the mass A of salt at time ris proportional to the square of the mass of salt present at time t. 17. Drag Race. Two drivers, Alison and Kevin, are par- ticipating in a drag race. Beginning from a standing start, they each proceed with a constant acceleration. Alison covers the last 1/4 of the distance in 3 sec- onds, whereas Kevin covers the last 1/3 of the dis- tance in 4 seconds. Who wins and by how much time? Historical Footer In 1630 Galileo formulated the brachistochrone problem (Bpáxiros-shortest, xpóros-time), that is, to determine a path down which a particle will fall from one given point to another in the shortest time. It was reproposed by John Bernoulli in 1696 and solved by him the following year. a differential equation is given along with the field or problem area in which it arises. Classify each as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ordinary differential equation, indicate whether the equation is linear or nonlinear 1. 4. dx² 2 gửi dr (mechanical vibrations, electrical circuits, seismology) (Hermite's equation, quantum-mechanical harmonic oscillator) 6. ax dy 2x +2y=0 dx +45 + 9x = 2 cos 3r dr + = 0 (Laplace's equation, potential theory, electricity, heat, aerodynamics) dy y(2-3x) dx x(1-3y) (competition between two species, ecology) dx 5. = k(4x)(1-x), where k is a constant dt (chemical reaction rates) + C, where C is a constant (brachistochrone problem, calculus of variations) dy 7. VI-y +2x- <=0 dx (Kidder's equation, flow of gases through a porous medium) 8. 9. 8 = x(1-x) dx dp = kp(P-p), where kand P are constants dr (logistic curve, epidemiology, economics) 11. 12. (deflection of beams) d'y 10. x + + xy =0 dx² dx (aerodynamics, stress analysis) a²N 1 aN + ar² rar aN ar (nuclear fission) + kN, where k is a constant dy -0.1(1-²) +9y=0 (van der Pol's equation, triode vacuum tube) In Problems 13-16, write a differential equation that fits the physical description. 13. The rate of change of the population p of bacteria at timer is proportional to the population at time 1. 14. The velocity at timer of a particle moving along a straight line is proportional to the fourth power of its position 1. 15. The rate of change in the temperature 7 of coffee at timer is proportional to the difference between the temperature M of the air at times and the tempera- ture of the coffee at time r. 16. The rate of change of the mass A of salt at time ris proportional to the square of the mass of salt present at time t. 17. Drag Race. Two drivers, Alison and Kevin, are par- ticipating in a drag race. Beginning from a standing start, they each proceed with a constant acceleration. Alison covers the last 1/4 of the distance in 3 sec- onds, whereas Kevin covers the last 1/3 of the dis- tance in 4 seconds. Who wins and by how much time? Historical Footer In 1630 Galileo formulated the brachistochrone problem (Bpáxiros-shortest, xpóros-time), that is, to determine a path down which a particle will fall from one given point to another in the shortest time. It was reproposed by John Bernoulli in 1696 and solved by him the following year.

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Here are the steps to solve each problem 1 fracd2ydx22xdydx2y0 This is an ODE since y is a function of x It is second order since there are two deriva... View the full answer