Consider the following differential equation. (1 + 5x^{2}) y′′ − 8xy′ − 6y = 0 |
(a) | If you were to look for a power series solution about
x_{0} = 0, i.e., of the form
then the recurrence formula for the coefficients would be given by c_{k+2} = g(k) c_{k} , k ≥ 2. Enter the function g(k) into the answer box below. |
(b) | Find the solution to the above differential equation with
initial conditions y(0) = 0 and
y′(0) = 9. (Note that this solution is a terminating power series.) |
(c) | Find the first three nonzero terms in the solution to the above differential equation with initial conditions y(0) = 7 and y′(0) = 0. |
Consider the following differential equation. (1 + 5x2) y′′ − 8xy′ − 6y = 0 (a)...
Consider the following differential equation. (1 + 5x2)y" – 8xY' – 6y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ τη x2 n=0 then the recurrence formula for the coefficients would be given by C +2 g(k) Cx. k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions (0) = 0 and...
Consider the following differential equation. (1 + 6x2)y" – 4xy' – 24y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form onth n=0 then the recurrence formula for the coefficients would be given by ck+2 = g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and y'(0) =...
cnrn Consider the following differential equation. (1 + 3x?) y" – 2xy' – 12y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σ n=0 00 then the recurrence formula for the coefficients would be given by Ck+2 g(k) Ck, k > 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and...
Consider the following initial value problem, (1 - z2)y"+zy' - 12y-0, (0)3, y' (0)-0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals (a) This differential equation has singular points at Note: You must use a semicolon here to separate your answers. (b) Since there is no singular point at z 0, you can find a normal power series solution for y(x about z0,i.e. m-0 As part of the solution process...
Consider the following differential equation to be solved using a power series. y" - y' = 0 Using the substitution y = į coxn, find an expression for Ck + 2 in terms of Ck + 1 for k = 0, 1, 2, .... k+2= + + + Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the s 4.3. Try to explain any differences between the two forms of the...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
(1 point) In this problem you will solve the differential equation (+7)y"+11xy' - y=0. x" for the differential equation will converge at least on the interval (-inf.-sqrt(7)] (1) Ey analyzing the singular paints of the differential equation, we know that a series solution of the form y = . (2) Substituting y = . *" into (x2+7y" + 11xy - y = 0, you get that Multiplying the coefficients in x through the sums E Reindex the sums Finally combine...
(1) Sok power series solution of the forma y(z)-Σ-oanz" to the differential equation: (a) (3 pts) Find recurrence relations for the coefficents, an (b) (4 pts) Use the recurrence relation to give the first three, n-zero terms of the power series solution to the initial value problem: y'-2xy = z, y(0) = 2 (c) (1 pt) Identify the solution as a common function (in closed form). (1) Sok power series solution of the forma y(z)-Σ-oanz" to the differential equation: (a)...
3. Consider the following differential equation 0o and a series solution to the differential equation of the form a" n-0 (a) Find the recurrence relations for the coefficients of the power series. 3 marks] (b) Determine the radius of convergence of the power series. l mar (c) Write the first eight terms of the series solution with the coefficients written in terms of ao and ai 2 marks] 3. Consider the following differential equation 0o and a series solution to...
differential equations Consider the following differential equation to be solved using a power series. y" + xy = 0 On Using the substitution y = cryn, find an expression for Ck + 2 in terms of Ck - 1 for k = 1, 2, 3... n = 0 Ck +2= + 6 Find two power series solutions of the given differential equation about the ordinary point x = 0. x3 O Y1 = 1 - xo and y2 = x...
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