Linear Equations

- Find all solutions to the following linear equations. You may need to leave answers in terms of
integrals on some of them.
(a) y

^{′}+ 2ty = e^{−t2 }(b) y^{′}− ty = e^{t}(c) y^{′}+ cosy = cos(d) y^{′}+ ty = sin(e) y^{′}+y =(f) y^{′}+ tany = cos(g) y^{′}− tany = sec(h) y^{′}− tany = sec^{2} - In the above linear equations find the solution to the initial value problems when yequals the following numbers.
(a) 1 (b) 2 (c) 3 (d) 4 (e) −2 (f) 12 (g) −3 (h) −2

- Solve the following initial value problem. ty
^{′}−y =,y= 2 . Would it make any sense to give the initial condition at t = 0? - Solve the following initial value problem. ty
^{′}+ y =,y= 2 . Would it make any sense to give the initial condition at t = 0? Hint: You need to remember that ∫dt = ln+ C. - Solve the following initial value problems.
- lny
^{′}+y = ln, y= 3 . - lny
^{′}−y = ln^{2}, y= 3 . - y
^{′}+ tany = cos^{3}, y= 4. - coshy
^{′}+ sinhy = sinh, y= −4

- ln
- You have the equation y
^{′}+ py = qwhere P^{′}= p. Give a formula for all solutions to this differential equation. - The height of an object at time t is y. It falls from an airplane at 30,000 feet which is traveling East at 500 miles per hour and is acted on by gravity which we will assume has acceleration equal to 32 feet per second squared and air resistance which we will suppose yields an acceleration equal to .1 times the speed of the falling object opposite to the direction of motion. If its initial velocity is in the direction of motion of the airplane, find a formula for the position of the object as a function of t in feet.
- Solve the following differential equations. Give the general solution.
- dx − xdy = 0
- ydx + dy = 0 Hint: You might look for x as a function of y.
- ydy = dx
- 2ydx =
- L+ Ri = E sin. Here L,R,E are positive constants. L symbolizes inductance and R resistance while i is the current.

- For compounding interest n times in one year which has interest rate r per year, the amount after t
years is given by A
_{0}^{tn}. Show thatthus giving the same conclusion as mentioned in the chapter.

- Consider the equation y
^{′}+ 2ty = t,y= 32 .76. Find lim_{t→∞}y. - Although the gas supply was shut off, the air in the building continued to circulate. When the gas was shut off, the temperature in the building was 70 and after five hours, the temperature had fallen to a chilly 50 degrees. If the outside temperature was at 10 degrees, what is the constant in Newton’s law of cooling?
- A radioactive substance decays according to how much is present. Thus the equation is A
^{′}= −kA. If after 40 years, there is 5∕6 of the amount initially there still present, what is the half life of this substance? - You have the following initial value problem y
^{′}+ y = sint, y= y_{0}. Letting y be the solution to this initial value problem, find a function uwhich does not depend on y_{0}and lim_{t→∞}= 0. - A pond which holds V cubic meters is being polluted at the rate of 10 + sinkg per year. The periodic source represents seasonal variability. The total volume of the lake is constant because it losesV cubic meters per year and gains the same. After a long time, what is the average amount of pollutant in this lake in a year?
Bernouli Equations

- Solve the following initial value problems involving Bernouli equations.
(a) y

^{′}+ 2xy = xy^{3}, y= 2 (b) y^{′}+ siny = siny^{2}, y= 1 (c) y^{′}+ 2y = x^{2}y^{3}, y= −1 (d) y^{′}− 2x^{3}y = x^{3}y^{−1}, y= 1 (e) y^{′}+ y = x^{2}y^{−2}, y= 1 (f) y^{′}+ x^{3}y = x^{3}y^{−2}, y= −1 - Consider y
^{′}= py − qy^{2},y=where p,q are positive and m > 1. Solve this Bernouli equation and also find lim_{t→∞}y. - Consider y
^{′}= 3y − y^{3}, y= 1 . Solve this Bernouli equation and find lim_{t→∞}y. - Find the solution to the Bernouli equation y
^{′}=y −y^{3}, y= 1 . Hint: You may have to leave the solution in terms of an integral. - Actually the drag force of a small object moving through the air is proportional not to the speed but
to the square of the speed. Thus a falling object would satisfy the following equation for downward
velocity. v
^{′}= g − kv^{2}. Here g is acceleration of gravity in whatever units are desired. Find lim_{t→∞}vin terms of g,k. Hint: Look at the equation. - A Riccati equation is like a Bernouli equation except you have an extra function added in. These are
of the form y
^{′}= a+ by + cy^{2}. If you have a solution, y_{1}, show that y= y_{1}+will be another solution provided v satisfies a suitable first order linear equation. Thus the set of all such y will involve a constant of integration and so can be regarded as a general solution to the Riccati equation. These equations result in a very natural way when you consider y^{′}= f(t,y) and approximate f(t,y) by fixing t and approximating the resulting function of y with a second order Taylor polynomial.Separable Equations

- Solve the following initial value problems involving separable equations. The ordered pair given is to
be included in the solution curve.
(a) x

^{2}dx +dy = 0,(b) xydx +dy = 0,(c) xydx +dy = 0,(d) ydx +xdy = 0,(e) 0 = cosdx + tandy,(f) xydx =dy,(g) xydx =dy, - Find all integral curves of the equation yxdx + e
^{−x2 }dy = 0. Graph several. - Find all integral curves of the equation yxdx + y
^{3}dy = 0. Graph several. - Give the integral curves to the equation v
^{′}= g − kv^{2}mentioned above where g is acceleration of gravity and k a positive constant. - You have a collection of hyperbolas x
^{2}− y^{2}= C where each choice of C leads to a different hyperbola. Find another collection of curves which intersect these at a right angle. Hint: Say you have f= C is one of these. If you are at a point where the relation defines y as a function of x, andis a point on one of these hyperbolas just mentioned, thenshould have a relation to the tangent line to x^{2}− y^{2}= C. Since the two curves are to be perpendicular, you should have the product of their slopes equal to −1. Thus= −1. - Generalize the above problem. Suppose you have a family of level curves f= C and you want another family of curves which is perpendicular to this family of curves at every point of intersection. Find a differential equation which will express this condition. Recall that two curves are perpendicular if the products of the slopes of the tangent lines to the two curves equals −1.
- Find and determine the stability of the equilibrium points for the following separable
equations.
(a) y

^{′}= y^{2}(b) y^{′}=(c) y^{′}= sin(d) y^{′}= cos(e) y^{′}= ln(f) y^{′}= e^{2y}− 1 (g) y^{′}= 1 − e^{y2 } - The force on an object of mass m acted on by the earth having mass M is given by
Newton’s formula kmM∕r
^{2}where k is the gravitation constant first calculated by Cavendish^{4}in 1798. Letting R be the radius of the earth and letting g denote the acceleration of gravity on the earth’s surface, show that kM = R^{2}g. Now suppose a large gun having its muzzle at the surface of the earth is fired away from the center of the earth such that the projectile has velocity v_{0}. Explain whywhere r is the distance to the surface of the earth and here v = v

the speed of the projectile at time t when it is at a distance of r from the surface of the earth. Next explain whyThe two variables are v and r. Separate the variables and find the solution to this differential equation given that the initial speed is v

_{0}as stated above. Show that the maximum distance from the surface of the earth is given byprovided that Rg >

v_{0}^{2}. What is the smallest value of v_{ 0}such that the projectile will leave the earth and never return? - The Grompertz equation is = ry ln. Find the solutions to this equation with initial condition y= y
_{0}. Also identify all equilibrium solutions and their stability. Also verify the inequality ry ln≥ ryfor y ∈. Explain why for a given initial condition y_{0}∈, the solution to the Grompertz equation should be at least as large as the solution to the logistic equation. - You have a population which satisfies the logistic equation y
^{′}= ryand the initial condition is y= αK where 0 < α < 1∕2. How long will it take for the population to double? - An equilibrium point is called semi-stable if it is stable from one side and not stable from the other.
Sketch the appearance of fnear y
_{0}if y_{0}is a semi-stable equilibrium point. Here f= 0 and the differential equation is y^{′}= f. - Consider the differential equation y
^{′}= a − y^{2}where a is a real number. Show that there are no equilibrium solutions if a < 0 but there are two of them if a > 0 and only one if a = 0. Discuss the stability of the two equilibrium points when a > 0. What about stability of equilibrium when a = 0? - Do exactly the same problem when y
^{′}= ay − y^{3}. This time show there are three equilibrium points when a > 0 and only one if a < 0. Discuss the stability of these points. - Do the same problem if y
^{′}= ay − y^{2}. These three problems illustrate something called bifurcation which is when the nature of the solutions changes dramatically when some parameter changes.Homogeneous Equations

- Find the solution curve to the following differential equations which contains the given
point.
- y
^{′}=^{2}, - y
^{′}= −, - y
^{′}=, - y
^{′}= −, - y
^{′}=, - y
^{′}=, - y
^{′}=,

- y
- Find the solution curve to the following ODEs which contains the given point.
- y
^{′}=, - y
^{′}=, - y
^{′}=, - y
^{′}=,

- y
- Find the solution curve to the following ODEs which contains the given point.
- −dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,
- −dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,.

- −
- Find all solutions to y
^{′}+ sin= 1 . Hint: You might need to leave the answer in terms of integrals. - Solve: x
^{2}dy +dx = 0, y= −1. - Solve: x
^{2}dy +dx = 0, y= −1. - Solve: x
^{2}dy +dx = 0, y= 1. - Solve: dx +dy = 0, y= −2.
Exact Equations and Integrating Factor

- Find the solution curve to the following ODEs which contain the given point. First verify that the
equation is exact.
- dx + x
^{2}dy = 0, - dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,

- Find the solution curve to the following ODEs which contains the given point.
- dx +dy = 0,
- dx +dy = 0,
- dx +dy = 0,
- 3y
^{4}dx +dy = 0,