INTRODUCTION TO DIFFERENTIAL EQUATIONS 20 CHAPTER 1 In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IVP. 3y2/3, y(0) = 0 15. y' 16. ху' — 2у, у(0) — 0 In Problems 17–24 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (xo, yo) in the region. dy dy = y2/3 17. dx 18. Vху dx dy 19. х dx dy 20. у3 х dx y²)y' = x² 22. (1 + y³)y' = x² 21. (4 23. (х2 + у?)у' у? 24. (у — х)у' %— у+x In Problems 25–28 determine whether Theorem 1.2.1 guarantees that the differential equation y' = Vy2 – 9 possesses a unique solution through the given point. 25. (1, 4) 26. (5, 3) 28. (–1, 1) 27. (2, –3) 29. (a) By inspection find a one-parameter family of solutions of the differential equation xy' = y. Verify that each member of the family is a solution of the initial-value problem xy' = y, y(0) = 0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy' = y would have a unique solution through a point (xo, yo) in R. (c) Verify that the piecewise-defined function |0, х<0 y = satisfies the condition y(0) function is also a solution of the initial-value problem in = 0. Determine whether this part (a). 30. (a) Verify that y = tan (x + c) is a one-parameter family of solutions of the differential equation y' = 1 + y². (b) Since f(x, y) erywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem v' = 1 + v², v(0) = 0. Even though xn = 0 is in = 1 + y? and af/əy = 2y are continuous ev-
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
#17
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 4 images