A( x) B( x) C( x) y( x) 0, A( x) 0

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3.1 Lexicon Revisited The nonhomogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x) F( x), A( x) dx dx The homogeneous nd Order ODE has the form: d y dy A( x) B( x) C( x) y( x), A( x) dx dx {3.1.3} {3.1.4} Eq {3.1.4} is said to be the homogeneous ODE associated with the nonhomogeneous ODE in Eq {3.1.3}. Common forms of nonhomogeneous and associated homogeneous nd Order ODEs are: d y dy p( x) q( x) y( x) f ( x) dx dx d y dy p( x) q( x) y( x) dx dx {3.1.5} {3.1.6} 3 October 18 1 Kidoguchi, Kenneth

Nonhomogeneous nd Order ODE with Constant Coefficients Consider the nonhomogeneous ODE: d d a y( x) a1 y( x) a y( x) f ( x) {3.5.1} dx dx with constant coefficients a, a 1, and a. Its associated homogeneous equation is: d d a y( x) a 1 y( x) a y( x) {3.5.} dx dx Let L be the linear polynomial operator: {3.5.1} and {3.5.} may be written. d d L a a a dx dx 1. Then Ly f ( x) {3.5.1a} Ly {3.5.a} 3 October 18 Kidoguchi, Kenneth

Theorem: General Solutions for Nonhomogeneous Equations If y p is any particular solution of the nonhomogeneous equation y'' + py' + qy = f(x) and y c be a general solution of its associated homogeneous equation, then y G = y p + y c is a general solution of the given nonhomogeneous equation. 1.5 Linear First-Order Equations, #31 Revisited P( x) dx a. Show that y Ce is a general solution of y + P(x) y(x) =. c P( x) dx P( x) dx b. Show that yp e Q( x) e dx is a particular solution of y + P(x) y(x) = Q(x). c. Suppose that y c (x) is any general solution y + P(x) y(x) = and y p (x) is any particular solution of y + P(x) y(x) = Q(x). Show that y c (x) + y p (x) is a general solution of y + P(x) y(x) = Q(x). 3 October 18 3 Kidoguchi, Kenneth

1.5 Linear First-Order Equations, #31 Revisited P( x) dx a) Show that y Ce is a general solution of y + P(x) y(x) =. c 3 October 18 4 Kidoguchi, Kenneth

1.5 Linear First-Order Equations, #31 Revisited P( x) dx P( x) dx b) Show that yp e Q( x) e dx is a particular solution of y + P(x) y(x) = Q(x). 3 October 18 5 Kidoguchi, Kenneth

1.5 Linear First-Order Equations, #31 Revisited c) Suppose that y c (x) is any general solution y + P(x) y(x) = and y p (x) is any particular solution of y + P(x) y(x) = Q(x). Show that y c (x) + y p (x) is a general solution of y + P(x) y(x) = Q(x). 3 October 18 6 Kidoguchi, Kenneth

PS3.1 #7 & 8 Revisited 7. Let y p be a particular solution of the nonhomogeneous equation y'' + py' + qy = f(x) and let y c be a solution of its associated homogeneous equation. Show that y = y p + y c is a solution to the given nonhomogeneous equation. 8. With y p = 1 and y c = c 1 cos(x) + c sin(x) in the notation of problem 7, find a solution of y'' + y = 1 satisfying the initial conditions y() = y'() = -1. 3 October 18 7 Kidoguchi, Kenneth

The Method of Undetermined Coefficients If y p is any particular solution of the nonhomogeneous equation Ly = y'' + py' + qy = f(x) and y c be a general solution of its associated homogeneous equation, Ly =, then y G = y p + y c is a general solution of the given nonhomogeneous equation. Rule 1: Suppose that no term appearing in f(x) or in any of its derivatives satisfies the associated homogeneous equation Ly =. Then take as a trial solution for y p a linear combination of all linearly independent such terms and their derivatives. Then determine the coefficients by substitution of this trial; solution into the nonhomogeneous equation Ly = f(x). 3 October 18 8 Kidoguchi, Kenneth

r Uncle Archimedes Consider a floating cylindrical buoy of radius r, height h, and uniform density r <.5 g/cm 3. Assume that the density of water is r = 1 g/cm 3. The buoy is initially suspended at rest with its bottom at the top surface of the water (x = ) and is released at time t =, x h Thereafter it is acted on by two forces: a downward gravitational force equal to its weight F r r hg and (by Archimedes) an upward force G equal to the weight of the displaced water, F r r xg where x(t) is the A depth of the bottom of the buoy beneath the surface at time t. Conclude that the buoy undergoes simple harmonic motion about its equilibrium position x rh/ r e with period T rh / g. Compute the period and amplitude of the motion if, r =.5 g/cm 3, h = cm, and g = 98 cm/s. 3 October 18 9 Kidoguchi, Kenneth

Uncle Archimedes Example Computation With the specified coordinate system, x measures the depth of the base of the buoy, so x = at the water's surface and increases downward. Then: F m g (buoy mass) g weight of buoy G B F m g (displace water mass) g Archimedes buoyant force A W 3 October 18 1 Kidoguchi, Kenneth

Uncle Archimedes Example Computation 3 October 18 11 Kidoguchi, Kenneth

Uncle Archimedes Example Computation 3 October 18 1 Kidoguchi, Kenneth

Uncle Archimedes With the specified coordinate system, x measures the depth of the base of the buoy, so x = at the water's surface and increases downward. Then: F mg r hg G F A r r r xg Net Force F F G B weight of buoy Archimedes buoyant force mx r r hg r r xg r r r r hx r hg r xg x r hg r rg r r r h r g g x r h r g x x g rh r r h x x x g, r g rh 3 October 18 13 Kidoguchi, Kenneth

Uncle Archimedes x x g x x, () () Let x c be a general solution to the homogeneous ODE: x ( t) a cos t a sin t c 1 x Let x p be a particular solution to the nonhomogeneous ODE: x () t p g grh rh r g r The general solution for the nonhomogeneous ODE: rh x( t) a cos 1 t a sin t r rh rh x() a a 1 1 v() a r r rh x( t) 1 cos t r r g rh x 3 October 18 14 Kidoguchi, Kenneth

Finding a Particular Solution, y p, Example 1 Given: y'' + y' + y = x, find y p (x), a particular solution to the ODE. 3 October 18 15 Kidoguchi, Kenneth

Finding a Particular Solution, y p, Example Given: y'' + 3y' + y = 4x +, find y p (x), a particular solution to the ODE. Select trial solution (i.e., guess!) : y p = Ax + B y Ax B 3A ( Ax B) 4x p y A A 4 A p y 3A B B p y x p 3 October 18 17 Kidoguchi, Kenneth

Finding y p when f(x) is Exponential: Example 3 Given: y'' + 3y' + y = 1 e x, find y p (x), a particular solution to the ODE. 3 October 18 18 Kidoguchi, Kenneth

Finding y p when f(x) is Exponential: Example 4 Given: y'' + 3y' + y = 1 e -x, find y p (x), a particular solution to the ODE. 3 October 18 Kidoguchi, Kenneth

3 October 18 1 Kidoguchi, Kenneth

Finding y p when f(x) is Exponential: Example 4 Given: y'' + 3y' + y = 1 e -x, find y p (x), a particular solution to the ODE. Select trial solution (i.e., guess!) : y p = A e -x. y Ae Ae 3Ae Ae 1e p x y Ae A 3A A 1 p x y Ae 1?????? p x x x x x Select another trial solution (i.e., guess again!) : y p = A xe -x. x x y Axe p Ae x 3(1 x) x 1e x x y A e xe A x 3 3x x 1 p x x x y A A 1 p e xe e x yp 1xe x 3 October 18 Kidoguchi, Kenneth

Finding y p when f(x) is Sinusoidal: Example 5 y'' + 3y' + y = 1cos(x). Find y p (x), a particular solution the ODE. 3 October 18 3 Kidoguchi, Kenneth

Finding y p when f(x) is Sinusoidal: Example 5 3 October 18 4 Kidoguchi, Kenneth

Finding y p when f(x) is Sinusoidal: Example 6 y'' + 3y' + y = 1sin(x). Find y p (x), a particular solution to the ODE. 3 October 18 7 Kidoguchi, Kenneth

The General Solution Recipe If y p is any particular solution of the nonhomogeneous equation Ly = y'' + py' + qy = f(x) and y c be a general solution of its associated homogeneous equation, Ly =, then y G = y p + y c is a general solution of the given nonhomogeneous equation. Given: Ly = y'' + py' + qy = f(x) Step 1: Find y c (x), the general solution of Ly =. Step : Find y p (x), a particular solution of Ly = f(x). Step 3: Find the general solution of Ly = f(x), y G (x), =y p (x) + y c (x) Step 4: Find the solution to the initial value problem, by solving for the constants in y G (x) that satisfy the specified initial conditions (ICs). 3 October 18 8 Kidoguchi, Kenneth

Given: 3.5: Nonhomogeneous Equations and Undetermined Coefficients Computation of y G: Example 5 y'' 3 y' y 1e x, find y G (x) a general solution to the ODE. Step 1: Find y c (x), the general solution of Ly =. rx y '' 3 y ' y assume y e 3 9 8 3 1, 4 4 r r r r r 1 x y ( x) c e c e c 1 Step : Find y p (x), a particular solution of Ly = f(x). 3 October 18 9 Kidoguchi, Kenneth x From previous example: y p = e x Step 3: Find the general solution of Ly = f(x), y G (x), =y p (x) + y c (x) y x c e c e e ( ) x x x G 1 Step 4: Find the solution to the initial value problem, by solving for the constants in y G (x) that satisfy the specified initial conditions (ICs).

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