Verify that y(t) = A cos(t ) is a solution to the differential equation (1.1.21), where

For Problems 14 determine the order of the differential equation.d2 y dx2 + x dy dx + y = ex .

For Problems 14 determine the order of the differential equation.dy dx 3 + y2 = sin x.

For Problems 14 determine the order of the differential equation.y + xy + ex y = y.

For Problems 14 determine the order of the differential equation.in(y) + x2 y + xy = ln x.

Verify that, for t > 0, y(t) = ln t is a solution to the differential equation 2 dy dt 3 = d 3 y dt3 . 6

Verify that y(x) = x/(x + 1) is a solution to the differential equation y + d2 y dx2 = dy dx + x3 + 2x2 3 (1 + x)3

Verify that y(x) = ex sin x is a solution to the differential equation

By writing Equation (1.1.7) in the form 1 T Tm dT dt = k and using u1 du dt = d dt (ln u), derive (1.1.8).

A glass of water whose temperature is 50F is taken outside at noon on a day whose temperature is constant at 70F. If the waters temperature is 55F at 2 p.m., do you expect the waters temperature to reach 60F before 4 p.m. or after 4 p.m.? Use Newtons law of cooling to explain your answer.

On a cold winter day (10F), an object is brought outside from a 70F room. If it takes 40 minutes for the object to cool from 70F to 30F, did it take more or less than 20 minutes for the object to reach 50F ? Use Newtons law of cooling to explain your answer.

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.x2 + 9y2 = c.

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.y = cx2.

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.y = c/x.

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.y = cx5.

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.y = cex .

For Problems 1116, find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.y2 = 2x + c.

For Problems 1720, m denotes a fixed nonzero constant, and c is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.y = cxm.

For Problems 1720, m denotes a fixed nonzero constant, and c is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.y = mx + c.

For Problems 1720, m denotes a fixed nonzero constant, and c is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.y2 = mx + c.

For Problems 1720, m denotes a fixed nonzero constant, and c is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.9. y2 = mx + c.

Consider the family of circles x2 + y2 = 2cx. Show that the differential equation for determining the family of orthogonal trajectories is dy dx = 2xy x2 y2 .

We call a coordinate system (u, v) orthogonal if its coordinate curves (the two families of curves u = constant and v = constant) are orthogonal trajectories (for example, a Cartesian coordinate system or a polar coordinate system). Let(u, v) be orthogonal coordinates, where u = x2+2y2, and x and y are Cartesian coordinates. Find the Cartesian equation of the v-coordinate curves, and sketch the (u, v) coordinate system.

Any curve with the property that whenever it intersects a curve of a given family it does so at an angle a = /2 is called an oblique trajectory of the given family. (See Figure 1.1.7.) Let m1 (equal to tan a1) denote the slope of the required family at the point (x, y), and let m2 (equal to tan a2) denote the slope of the given family. Show that m1 = m2 tan a 1 + m2 tan a . [Hint: From Figure 1.1.7, tan a1 = tan(a2a)]. Thus, the equation of the family of oblique trajectories is obtained by solving dy dx = m2 tan a 1 + m2 tan a . Curve of given family m1 5 tan a1 5 slope of required family m2 5 tan a2 5 slope of given family a1 a2 a Curve of required family Figure 1.1.7: Oblique trajectories intersect at an angle a.

An object is released from rest at a height of 100 meters above the ground. Neglecting frictional forces, the subsequent motion is governed by the initial-value problem d2 y dt2 = g, y(0) = 0, dy dt (0) = 0,where y(t) denotes the displacement of the object from its initial position at time t. Solve this initial-value problem and use your solution to determine the time when the object hits the ground.

A five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, the subsequent motion is governed by the differential equation d2 y dt2 = g. If the object hits the ground 8 seconds after the boy releases it, find (a) the time when the tennis ball reaches its maximum height. (b) the maximum height of the tennis ball.

A pyrotechnic rocket is to be launched vertically upwards from the ground. For optimal viewing, the rocket should reach a maximum height of 90 meters above the ground. Ignore frictional forces. (a) How fast must the rocket be launched in order to achieve optimal viewing? (b) Assuming the rocket is launched with the speed determined in part (a), how long after the rocket is launched will it reach its maximum height?

Repeat Problem 26 under the assumption that the rocket is launched from a platform five meters above the ground.

An object that is initially thrown vertically upward with a speed of 2 meters/second from a height of h meters takes 10 seconds to reach the ground. Set up and solve the initial-value problem that governs the motion of the object, and determine h.

An object that is released from a height h meters above the ground with a vertical velocity of v0 meters/second hits the ground after t0 seconds. Neglecting frictional forces, set up and solve the initial-value problem governing the motion, and use your solution to show that v0 = 1 2t0 (2h gt2 0 ).

Verify that y(t) = A cos(t ) is a solution to the differential equation (1.1.21), where A and are nonzero constants. Determine the constants A and (with || < radians) in the particular case when theinitial conditions are y(0) = a, dy dt (0) = 0.

Verify that y(t) = c1 cos t + c2 sin t is a solution to the differential equation (1.1.21). Show that the amplitude of the motion is A = c2 1 + c2 2.

A heron has a birth mass of 3 g, and when fully grown its mass is 2700 g. Using equation (1.1.26) with a = 1.5 determine the mass of the heron after 30 days.

A rat has a birth mass of 8 g, and when fully grown its mass is 280 g. Using equation (1.1.26) with a = 0.25 determine how many days it will take for the rat to reach 75% of its fully grown size.