Please answer (i) (ii) and (iii)
Please answer (i) (ii) and (iii) 5. For each of the following linear homogeneous ODE, do...
just focus on A,B,D 1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
IGNORE (i) (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the “method of undetermined series coefficients”. (iii) The underlying idea behind the method of undetermined coefficients is a conjecture about the form of a particular solution that is motivated by the right-hand side of the equation. The method of undetermined coefficients is limited to second-order linear ODEs with constant coefficients and the right-hand side of the ODE cannot be an...
Consider the following statements. (i) Given a second-order linear ODE, the method of variation of parameters gives a particular solution in terms of an integral provided y1 and y2 can be found. (ii) The Laplace Transform is an integral transform that turns the problem of solving constant coefficient ODEs into an algebraic problem. This transform is particularly useful when it comes to studying problems arising in applications where the forcing function in the ODE is piece-wise continuous but not necessarily...
Consider the following statements. (i) A Taylor series is a power series that gives the expansion of a function around a point a. Convergence of such series is fully understood by means of the ratio test. (ii) We must rethink what we mean by solving y′′ + y′ − y = { cos(x + 42) x ≠ 1 0 x = 1 before trying to compute a solution defined on an interval containing x = 1. (iii) Most of the...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
A.9. First-order linear non-homogeneous ODEs having one dependent variable are of the form dy + P(x)y = f(x). Beginning with yp = uyż, where yı = e-SP(x)dx and is thus a solution to Y + P(x)y = 0, and given that the general solution y = cyı + Yp, use variation of parameters to derive the formula for the general solution to first-order linear non-homogeneous ODES: dx y = e-SP(x)dx (S eS P(x)dx f(x)dx + c). You may use the...
I need help solving these problems 1. Suppose that y= a (x-1)" is the power series solution of the following initial value problem. x-y+2y=0; y(t) = -2, y(1)=1 Find the value of az. 2. Suppose that y=0(x) is the solution of the following initial value problem. y" + xy - (sinx)y=0; y(0)=1, 7(0) = 3 Find the value of (0) 3. Let p be the radius of convergence for the Taylor series of the following rational function centered at the...
Please show all work and steps! Would like to learn how! Given a second order linear homogeneous differential equation a2(x)y" + a1(x)y' + 20 (x)y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions Yı, Y2. But there are times when only one function, call it Yı, is available and we would like to find a second linearly independent solution. We can find Y2 using the method of reduction of order....
Consider the following statements. (i) The Laplace Transform of 11tet2 cos(et2) is well-defined for some values of s. (ii) The Laplace Transform is an integral transform that turns the problem of solving constant coefficient ODEs into an algebraic problem. This transform is particularly useful when it comes to studying problems arising in applications where the forcing function in the ODE is piece-wise continuous but not necessarily continuous, or when it comes to studying some Volterra equations and integro-differential equations. (iii)...
i. 1. Answer each of the following For each of the following differential equations, state the order of the equation and state whether it is linear or nonlinear. If the differential equation is linear, state whether it is homogeneous or nonhomogeneous dy + + xy = sin x dx 2 a. dx2 b. x6y(5) – x2y'" – (cos x )y – ex = 0 ii. Find the value(s) of m so that the function y = xº, x 0 is...