Calculus 3 5: power series

Infinite Series and Differential Equations Nguyen Thieu Huy Hanoi University of Science and Technology Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 1/9 Differential Equations: Motivations Falling Body Problems Vertically falling body of mass m with unknown velocity v : Two forces: 1) Gravitation mg ; 2) Air resistance: −kv , where k: positive constant Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 2/9 Differential Equations: Motivations Falling Body Problems Vertically falling body of mass m with unknown velocity v : Two forces: 1) Gravitation mg ; 2) Air resistance: −kv , where k: positive constant dv v Newton’s law: Ftotal = ma ⇔ mg − kv = m dv dt ⇔ dt + k m = g Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 2/9 Electrical Circuits RL-circuit consisting of a resistance R, inductor L, electromotive force E Let find the current I : dI Kirchoff’s Voltage Law: VR + VL = E Also, VR = RI ; VL = L dt Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 3/9 Electrical Circuits RL-circuit consisting of a resistance R, inductor L, electromotive force E Let find the current I : dI Kirchoff’s Voltage Law: VR + VL = E Also, VR = RI ; VL = L dt dI dI RI + L = E ⇔ + RL I = E dt dt Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 3/9 Electrical Circuits RL-circuit consisting of a resistance R, inductor L, electromotive force E Let find the current I : dI Kirchoff’s Voltage Law: VR + VL = E Also, VR = RI ; VL = L dt dI dI RI + L = E ⇔ + RL I = E dt dt 2.1 Definition A differential equation is an equation involving an unknown function and its derivatives Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 3/9 Basic Notions Unknown function y : • A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 4/9 Basic Notions Unknown function y : • A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable • If the unknown function depends on two or more ind variables, then the differential equation is a partial differential equation Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 4/9 Basic Notions Unknown function y : • A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable • If the unknown function depends on two or more ind variables, then the differential equation is a partial differential equation • The order of a differential equation is the order of the highest derivative appearing in the equation Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 4/9 Basic Notions Unknown function y : • A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable • If the unknown function depends on two or more ind variables, then the differential equation is a partial differential equation • The order of a differential equation is the order of the highest derivative appearing in the equation Abbreviation: DE: Differential Equation Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 4/9 Example: The problem y ” + 2y = x; y (π) = 1, y (π) = is an initial value problem, because the two subsidiary conditions are both given at x = π Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 6/9 Example: The problem y ” + 2y = x; y (π) = 1, y (π) = is an initial value problem, because the two subsidiary conditions are both given at x = π The problem y ” + 2y = x; y (0) = 1, y (1) = is a boundary-value problem, because the two subsidiary conditions are given at x = and x = Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 6/9 Example: The problem y ” + 2y = x; y (π) = 1, y (π) = is an initial value problem, because the two subsidiary conditions are both given at x = π The problem y ” + 2y = x; y (0) = 1, y (1) = is a boundary-value problem, because the two subsidiary conditions are given at x = and x = 2.5 Standard and Differential Forms Standard form for a first-order DE in the unknown function y (x) is y = f (x, y ) Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq (1) 6/9 Example: The problem y ” + 2y = x; y (π) = 1, y (π) = is an initial value problem, because the two subsidiary conditions are both given at x = π The problem y ” + 2y = x; y (0) = 1, y (1) = is a boundary-value problem, because the two subsidiary conditions are given at x = and x = 2.5 Standard and Differential Forms Standard form for a first-order DE in the unknown function y (x) is y = f (x, y ) (1) The right side of (1) can always be written as a quotient of two other M(x,y ) functions −M(x, y ) and N(x, y ) Then (1) becomes dy dx = − N(x,y ) ⇔ Differential form M(x, y )dx + N(x, y )dy = Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq (2) 6/9 FIRST-ORDER DIFFERENTIAL EQUATIONS Separable Equations 1.1 Definition The first-order separable differential equation has the form A(x)dx + B(y )dy = (3) 1.2 Solutions: The solution to (3) is A(x)dx + B(y )dy = c (4) where c represents an arbitrary constant Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 7/9 FIRST-ORDER DIFFERENTIAL EQUATIONS Separable Equations 1.1 Definition The first-order separable differential equation has the form A(x)dx + B(y )dy = (3) 1.2 Solutions: The solution to (3) is A(x)dx + B(y )dy = c (4) where c represents an arbitrary constant x +2 Example Solve the equation: dy dx = y Rewrite in the differential form (x + 2)dx − ydy = which is separable with A(x) = x + and B(y ) = −y Its solution is (x + 2)dx − ydy = c or 31 x + 2x − 12 y = c for any constant c Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 7/9 FIRST-ORDER DIFFERENTIAL EQUATIONS Separable Equations 1.1 Definition The first-order separable differential equation has the form A(x)dx + B(y )dy = (3) 1.2 Solutions: The solution to (3) is A(x)dx + B(y )dy = c (4) where c represents an arbitrary constant x +2 Example Solve the equation: dy dx = y Rewrite in the differential form (x + 2)dx − ydy = which is separable with A(x) = x + and B(y ) = −y Its solution is (x + 2)dx − ydy = c or 31 x + 2x − 12 y = c for any constant c Sometimes, it may not be algebraically possible to solve for y explicitly in terms of x In that case, the solution is left in implicit form Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 7/9 1.3 Solutions to the Initial-Value Problem: The solution to the initial-value problem A(x)dx + B(y )dy = 0; y (x0 ) = y0 can be obtained first by formula (4) and then substitute y (x0 ) = y0 to find x y constant c, or another way is using x0 A(t)dt + y0 B(s)ds = Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 8/9 1.3 Solutions to the Initial-Value Problem: The solution to the initial-value problem A(x)dx + B(y )dy = 0; y (x0 ) = y0 can be obtained first by formula (4) and then substitute y (x0 ) = y0 to find x y constant c, or another way is using x0 A(t)dt + y0 B(s)ds = Homogeneous Equations: 2.1 Definition: A DE in standard form dy = f (x, y ) dx is homogeneous if f (tx, ty ) = f (x, y ) for every real number t = Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq (5) 8/9 1.3 Solutions to the Initial-Value Problem: The solution to the initial-value problem A(x)dx + B(y )dy = 0; y (x0 ) = y0 can be obtained first by formula (4) and then substitute y (x0 ) = y0 to find x y constant c, or another way is using x0 A(t)dt + y0 B(s)ds = Homogeneous Equations: 2.1 Definition: A DE in standard form dy = f (x, y ) dx is homogeneous if f (tx, ty ) = f (x, y ) for every real number t = (5) Consider x = Then, f (x, y ) = f (x · 1, x · yx ) = f (1, yx ) := g ( yx ) for a function g depending only on the ratio yx Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 8/9 1.3 Solutions to the Initial-Value Problem: The solution to the initial-value problem A(x)dx + B(y )dy = 0; y (x0 ) = y0 can be obtained first by formula (4) and then substitute y (x0 ) = y0 to find x y constant c, or another way is using x0 A(t)dt + y0 B(s)ds = Homogeneous Equations: 2.1 Definition: A DE in standard form dy = f (x, y ) dx is homogeneous if f (tx, ty ) = f (x, y ) for every real number t = (5) Consider x = Then, f (x, y ) = f (x · 1, x · yx ) = f (1, yx ) := g ( yx ) for a function g depending only on the ratio yx 2.2 Solution: The homogeneous differential equation can be transformed into a separable equation by making the substitution: y = xv with its dv dv dv dx derivative: dy dx = v + x dx Then, v + x dx = g (v ) ⇔ g (v )−v = x for g (v ) = v , which is a separable DE and can be solved as previously For g (v ) = v it yields another solution of the form y = kx for any constant k Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 8/9 Exact equations 3.1 Definition A differential equation in differential form M(x, y )dx + N(x, y )dy = (6) is exact if there exists a function g (x, y ) such that dg (x, y ) = M(x, y )dx + N(x, y )dy Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 9/9 Exact equations 3.1 Definition A differential equation in differential form M(x, y )dx + N(x, y )dy = (6) is exact if there exists a function g (x, y ) such that dg (x, y ) = M(x, y )dx + N(x, y )dy Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 9/9 Exact equations 3.1 Definition A differential equation in differential form M(x, y )dx + N(x, y )dy = (6) is exact if there exists a function g (x, y ) such that dg (x, y ) = M(x, y )dx + N(x, y )dy 3.2 Test for exactness: If M(x, y ) and N(x, y ) are continuous functions and have continuous first partial derivatives on some rectangle of the xy -plane, ) ) then Equation (6) is exact if and only if ∂M(x,y = ∂N(x,y in this domain ∂y ∂x Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 9/9 Exact equations 3.1 Definition A differential equation in differential form M(x, y )dx + N(x, y )dy = (6) is exact if there exists a function g (x, y ) such that dg (x, y ) = M(x, y )dx + N(x, y )dy 3.2 Test for exactness: If M(x, y ) and N(x, y ) are continuous functions and have continuous first partial derivatives on some rectangle of the xy -plane, ) ) then Equation (6) is exact if and only if ∂M(x,y = ∂N(x,y in this domain ∂y ∂x 3.3 Solution: To solve Equation (6), first solve the equations ∂g (x,y ) = M(x, y ) ∂x for g (x, y ) ∂g (x,y ) = N(x, y ) ∂y The solution to (6) is g (x, y ) = c where c: arbitrary constant Nguyen Thieu Huy (HUST) Infinite Series and Diff Eq 9/9