a. Use Power Series to find the solution of the homogeneous DE given by + 3xy = 0.d²ydx²b. Start with the solution y(x)= a₁ + ax + a₂x² + a₂x³ + a²x¹ + açx³ +..., and calculate theexpansions for the first two terms on the LHS of the equation.c. Get the recurrence relations between the a's and find the first 5 terms in the solution.d. Suppose that the initial condition for y(x) is y(0) = 1 and y'(0) = 0. What will the series becomein this case?

Answered: a. Use Power Series to find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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H is a fixed number , AB=15 f(x) = { 0 -AB < x < 0 0 < x < AB H. Find the period of the function, calculate the coefficients in the Fourier series expansion, and mark which values n=1 are. а. an=0.637H; a0=0;bn=0 O b. an=0; a0=H;bn=0.637H O c. an=2.pi.H/11; a0=H/11;bn=0 O d. an=H; a0=0;bn=H е. an=0; a0=H;bn=0
The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
3. Find the recurrence relation for y" + 3ry – y = 0. Use the initial conditions y(0) = 2 and y (0) = 0 to find the values the first four non-zero coefficients in y = n=0
Find singular points of the differential equation 2x2y′′ − xy′ + (1 + x)y = 0, and solve it around that singular points using power series solutions. Find the first three nonzero terms in each of two solutions about the singular point. Solve the recurrence relation you find (i.e express an in terms of n) and finally show that solutions you find is converging. (Hint: Find the roots of indicial equation and construct the recurrence relation).
(a) Seek power series solutions of Airy differential equation y" – xy = 0 about x = 0; find the recurrence relation. (b) Find the first four terms in each of two solutions y1 and y2.
2) Suppose you have a differential equation that satisfies the conditions of Theorem 6.2.1- Existence of Power Series Solutions with center x, = 0. It is then guaranteed that a solution exists of the form y(x) = E-0 Cnx". The recurrence relation for the coefficients was calculated to be k 3 Ck+2 k = 1,2,3, .. 2(k + 2) k+1 2(k + 2)(k + 1) 3 C2 = -7C0 Find the solution y(x) to the initial-value problem if y(0) = 0 and y'(0) = 1. Give at least the first 3 nonzero terms of the solution.
In solving a differential equation by Frobenius method of series solution about x = 0, we obtain the following indicial equation and recurrence relation: 7(2r–1)=0 and 1 Cn+1 = %3D 6n +r+3n where n = 0,1,2,3,... Find the solution corresponding to the larger root of the indicial equation. Include the first three nonzero terms, use Co=I O A. Y = 1+x 4 x+... 133 O B. Y =x-1/2/ 4 3 +.. 133 O c.y =xl/2| 12 4 3 +... c. 4 2 O D. y = x1/2 1+x+ 133 ++.. O E.Y =x1/2 1+x+ 19 4 2 +...
2.3.9 Write down the first three terms of the Taylor series expansions of the functions dx (th) and (1-2h) dx dt dt about x(t). Use these two equations to eliminate d²x (t) and d³x dt³ dt from the Taylor series expansion of the function x(t + h) about x(t). Show that the resulting formula for x(t + h) is the third member of the Adams- Bashforth family, and hence confirm that this Adams-Bashforth method is a third-order method. €
1. Given the function f(x) = −2x³ + 12x²-20x+8,5 Using the Taylor series of order zero, one, two, and three, estimate the function at the point Xi+1= 0,5 based on the function value x;= 0.
2.3.9 Write down the first four terms of the Taylor series expansion of the function x(t-h) about x(t), and the first three terms of the expansion of the function dx dt (t-h) about x(t). Use these two equations to eliminate d²x (t) and d³x dt³ (t) dt² from the Taylor series expansion of the function x(t + h) about x(t). Show that the resulting formula is Xn+1 = -4X + 5X-1 + h(4F + 2F-1) + 0(ht) Show that this method is a linear combination of the second-order Adams-Bashforth method and the central difference method (that is, the scheme based on (2.9)). What do you think, in view of this, might be its disadvantages?
Find the recurrence relation of the coefficients for the two linearly independent power series solutions for the differential equation: y" + 2xy' + 4y = 0
Consider the differential equation (x2−1)y′′+[1−(a+b)]xy′+aby=0, (11.7.9) where a and b are constants.

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