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Section 1.1 Differential Equations & Mathematical Models Differential Equations – “Equations with derivatives in them.” Examples: 1. dy xy e x dx 2. 3x y 4x 2 1 y 1 3. y″ – 2 = 3x ln(x) + y2 What is a solution to a differential equation? A differential equation will often have infinitely many solutions. For example, here are some of the many solutions of dy dx 1. y = x2 2. y = x2 + e–x 3. y = x2 + 4e–x 4. y = x2 + 31429674e–x : : “Family of solutions” y x 2x 2 Calculus Review: If dy dx f (x) then y f ( x ) dx A differential equation will usually have infinitely many solutions, but there times when a differential equation will have only one solution or no solutions. Example: (y ′)2 + y2 = –1 Some differential equations will have solutions, but unfortunately we can't write them down (in terms of our elementary functions). Example: y e x 2 General Solutions vs. Particular Solutions 1. Solve dy 3x 2 3x 2 3x 2 dx 2. Solve 3. Solve dy dx dy dx if y = 5 when x = 1. , y(1) = 5. Initial Value Problem – consists of a differential equation along with an initial condition y(xo) = yo Definition: Order The order of a differential equation is the order of the highest derivative appearing in it. Expressing differential equations: Often we will be able to express 1st order differential equations as dy dx f ( x , y ). Expressing differential equations: We will always be able to express. . . . 1st order differential equations in the form F(x, y, y′) = 0 2nd order differential equations in the form F(x, y, y′, y″) = 0 : nth order differential equations in the form F(x, y, y′, y″, y″′, . . . . , y(n)) = 0 Definition: Solution to a Differential Equation A function u(x) is a solution to the differential equation F(x, y, y′, y″, . . , y(n)) = 0 on an interval J if u, u′, u″, . . . , u(n) exist on J and F(x, u, u′, u″, . . . , u(n)) = 0 for all x on J. Ex. 1 1 (a) Show that y(x) = 1/x is a solution to 3 xy 2 y on the interval [1, 20]. x Ex. 1 1 (b) Show that y(x) = 1/x is not a solution to 3 xy 2 y on the interval [-20, 20]. x Ex. 2 (a) Show that y1(x) = sin(x) is a solution to (y ′ )2 + y2 = 1 (b) Show that y2(x) = cos(x) is a solution to (y ′ )2 + y2 = 1 Partial Derivatives Ordinary Differential Equations vs. Partial Differential Equations Section 1.2 Integrals as General & Particular Solutions Ex. 1 Solve dy dx co s x Ex. 2 Solve x dy dx 1 2x 1 2 Ex. 3 Solve x dy dx 1 2 x 1, 2 y (1) 3 Position - Velocity – Acceleration s(t) = position s′ (t) = velocity Force = Mass x Acceleration s″ (t) = acceleration Ex. 5 A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)? Ex. 5 A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)? Section 1.3 Slope Fields & Solution Curves Slope field for dy dx co s x Slope field for dy dx 2x Ex. 1 Sketch the slope field for y′ = –x Ex. 2 Sketch the slope field for y′ = x2 + y2 2 2 Ex. 3 Examine some solution curves of y x dy dx 0. On the following slope field, draw the solution curve which satisfies the initial condition of. . . . . (a) y(2) = –1 (b) y(–1) = 3 (c) y(0) = 0 (d) y(0) = 1 Calculus Review (definition of continuity): f (x) is continuous at xo if lim f ( x ) f ( x o ) x xo Calculus Review (definition of continuity): f (x) is continuous at xo if lim f ( x ) f ( x o ) x xo f (x, y) is continuous at (xo, yo) if lim x , y xo , y o f x , y f xo , yo Theorem I: Existence & Uniqueness of Solutions Suppose that f (x, y) is continuous on some rectangle in the xy-plane containing the point (xo, yo) in its interior and that the partial derivative fy is continuous on that rectangle. Then the initial value problem dydx f ( x , y ); y ( x o ) y o , has a unique solution on some open interval Jo containing the point xo. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (a) dy dx x 3y , 2 5 y (2) 4 Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (b) y 2 x 2 d y 0, dx y (0 ) 1 Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (c) y x 2 2 dy dx 0, y (1) 0 Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (d) dy dx 1 x y , y (2) 2 Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (e) dy dx 1 x y , y (3) 5 Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (f) dy dx 7 x 7 y, y (3) 0 Section 1.4 Separable Equations & Applications Definition: Separable Differential Equation A first order differential equation dy dx f ( x, y ) is said to be separable if f (x, y) can be written as a product of a function of x and a function of y (i.e. dy dx g ( x)h( y ) ). Definition: Separable Differential Equation A first order differential equation dy dx f ( x, y ) is said to be separable if f (x, y) can be written as a product of a function of x and a function of y (i.e. dy g ( x)h( y ) ). dx Examples: 1. 2. 3. dy dx 3 x 1 7 y 2 y 3 2 y e 1 2 sin x x dx y 5 dy dy dx x 3x 6 2 y y 10 5 dy g (x) To solve a separable differentiable equation of the form dx h( y) we proceed as follows: dy g (x) To solve a separable differentiable equation of the form dx h( y) we proceed as follows: dy dx g (x) h( y) h(y) dy = g(x) dx dy g (x) To solve a separable differentiable equation of the form dx h( y) we proceed as follows: dy dx g (x) h( y) h(y) dy = g(x) dx h ( y ) dy g ( x ) dx (Then integrate both sides and solve for y, if this is possible.) Ex. 1 Solve dy dx 4x y 3 2 Ex. 2 Solve dy dx x y Justification for why this method for solving separable differentiable equations actually works. Ex. 3 Solve dy dx 6 xy , y (0 ) 7 Ex. 4 Solve dy dx x y Review of general solutions and particular solutions. Definition: Singular Solution A particular solution to a first order differential equation is said to be a singular solution if it does not come from the general solution. Ex. 5 Solve dy dx 6 x y 1 2 /3 Ex. 5 Solve dy dx 6 x y 1 2 /3 In general: If we have the differential equation dy dx g ( x)h( y ) , and h(y) has a zero of yo then the function y(x) = yo will be a singular solution. Applications If y changes at a rate proportional to y then Radioactive material & half-lives dy dx ky (for some constant k). Ex. 6 A radioactive substance has a half-life of 5 years. Initially there are 128 grams of this substance. How much remains after t years? Ex. 7 The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be until the region is again habitable? Carbon dating Ex. 8 Carbon extracted from an ancient skull contained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull? Newton's law of cooling (heating) According to Newton's law of cooling, the time rate of change of the temperature T of a body immersed in a medium of constant temperature A is proportional to the difference T – A . That is: dT dt k T A . Ex. 9 A cake is removed from an oven at 210° F and left to cool at a room temperature, which is 70° F. After 30 min the temperature of the cake is 140° F. When will it be 100° F? Section 1.5 Linear First-Order Equations Definition: Linear Differential Equation (First Order) A first order differential equation is linear if there are functions P(x) and Q(x) so that dy dx Examples: 1. y′ + sin(x) y = ex 2. y′ – sin(x) y = ex 3. y′ = 3x2y + x3 – 4x + 1 4. cos(x) y′ – sec(x) y = x2 P (x) y Q (x) Definition: Linear Differential Equation (First Order) A first order differential equation is linear if there are functions P(x) and Q(x) so that dy dx Integrating Factor: P ( x ) dx e P (x) y Q (x) Steps you MUST show when solving a 1st order linear differential equation: 1. Put the differential equation in the form: dy P (x) y Q (x) dx 2. Compute μ. 3. Multiply μ on both sides of the differential equation to obtain dy dx 4. Write this as (μy)′ = μ Q(x) 5. Solve this last differential equation via integration. P ( x) y Q ( x) Ex. 1 Solve: dy dx x y Why isn't there a “+C” in the integrating factor μ? Ex. 2 Solve: x y′ + 2y = 10x3; y(1) = 5 Reminder: (Theorem I from section 1.3) Suppose that f (x, y) is continuous on some rectangle in the xy-plane containing the point (xo, yo) in its interior and that the partial derivative fy is continuous on that rectangle. Then the initial value problem dy dx f ( x , y ); y ( xo ) yo , has a unique solution on some open interval Jo containing the point xo. Theorem I If the functions P(x) and Q(x) are continuous on the open interval J containing the point xo, then the initial value problem dy P ( x ) y Q ( x ); dx has a unique solution y(x) on J . y ( xo ) yo Ex. 3 Solve: y y x 2; y (2) 4 Reminder of a result from calculus: dx dy 1 dy dx Applications Mixture Problems Solutes, Solvents, & Solutions Q = amount of solute, V = volume of solution ri = rate in ro = rate out, ci = concentration in, co = concentration out Ex. 4 Consider a large tank holding 1000 L of water into which a brine solution of salt begins to flow at a constant rate of 6 L/min. The solution inside the tank is kept well stirred and is flowing out of the tank at a rate of 6 L/min. If the concentration of salt in the brine entering the tank is 1 kg/L, determine when the concentration of salt in the tank will reach 0.5 kg/L. Ex. 5 For the mixture problem described in example 4, assume now that the brine leaves the tank at a rate of 5 L/min instead of 6 L/min and assume that the tank starts out with a concentration of 0.1 kg/L (everything else stays the same as it was in example 4 though). Determine the concentration of salt in the tank as a function of time. Ex. 6 A swimming pool whose volume is 10,000 gallons contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min, and the pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hr? When will the pool be 0.002% chlorine? Section 1.6 Substitution Methods & Exact Equations Ex. 1 Solve dy dx 9 x y 4 2 If a differential equation can be written as dy f ( A x B y C ), dx let u = Ax + By + C and the resulting differential equation in terms of u and x will be separable. Definition: First Order Homogenous Differential Equation dy y f A 1st order differential equation that can be expressed as dx x is said to be homogenous. Examples: 1. 2 y y 3 7 dx x x dy 2. y 5y y sin 2 x x 3. y 5x y sin 2 y x 4. x 2 2 dy dx x y 2 2 If a differential equation can be written as y f , dx x dy let u y x and the resulting differential equation in terms of u and x will be separable. Ex. 2 Solve x 2 dy dx xy x e 2 y/x Here's why a 1st order homogenous differential equation can be turned into a y separable differential equation by the substitution u . x Here's why a 1st order homogenous differential equation can be turned into a y separable differential equation by the substitution u . x y f dx x dy Here's why a 1st order homogenous differential equation can be turned into a y separable differential equation by the substitution u . x y f dx x dy d dx ux f (u ) Here's why a 1st order homogenous differential equation can be turned into a y separable differential equation by the substitution u . x y f dx x dy d dx x ux du dx f (u ) u f (u ) Here's why a 1st order homogenous differential equation can be turned into a y separable differential equation by the substitution u . x y f dx x dy d dx x ux du f (u ) u f (u ) dx du dx f (u ) u x Definition: Bernoulli Differential Equation dy n P (x) y Q (x) y order differential equation that can be expressed as dx is said to be a Bernoulli differential equation. A 1st Examples: 1. dy dx 3 xy sin x y 1 3x 8 2. y 4e y 3. x y 5 tan x y sinh x y x 2 5 2x y 10 5 Ex. 3 Solve dy dx 2 x y (8 x 5) y 3 4 Ex. 3 Solve dy dx 2 x y (8 x 5) y 3 4 Definition: Exact Differential Equation A 1st order differential equation that can be expressed as M ( x , y ) N ( x , y ) dy dx with is said to be an exact differential equation. 0 Exact Differential Equation M ( x, y ) N ( x, y ) dy 0 dx M(x, y) dx + N(x, y) dy = 0 Exact Differential Equation Solving M(x, y) dx + N(x, y) dy = 0 Ex. 4 Solve (4x – y) dx + (6y – x) dy = 0 2 xy 3 x 2 Ex. 5 Solve dy dx 2x y 4 y 2 2 3 Section 1.8 Acceleration-Velocity Models Ex. 1 Suppose that an object is a distance of ho from the surface of the earth when it is given an initial velocity of vo. Determine s(t), the position of the object expressed as a function of time. (Assume that the only force acting on the object is due to gravity. Assume a constant acceleration due to gravity of g). Ex. 2 Suppose that an object is a distance of ho from the surface of the earth when it is given an initial velocity of vo. Determine s(t), the position of the object expressed as a function of time, this time including air resistance. (Use g for the acceleration due to gravity and assume that the force caused by the air resistance is proportional to the velocity.) Ex. 3 An object of mass 3 kg is released from rest 500 m above the ground and allowed to fall under the influence of gravity. Assume the force due to air resistance is proportional to the velocity of the object with a drag coefficient of 4/3 sec–1. Determine when the object will strike the ground. Newton's Law of Gravitation – The gravitational force of attraction between two point masses M and m located at a distance r apart is given by F GMm r 2 . Ex. 4 A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 . At what height above the lunar surface should the retro rockets be activated to ensure a soft touchdown (velocity = 0 at impact)?