63. Prove that (-1)k 2(x) = L 2k+2 k! (k + 3)! 00 24+2 k=0 is a solution of the Bessel differential equation of order 2: x?y" +xy' + (x2- 4)y = 0

Q: (b) Use the substitution x = e' to transform the Cauchy-Euler equation x²y" – 4xy' + 6y = 3x² +…

A:

Q: Verify that the indicated function y = g(x) is an explicit solution of the given first-order…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: 1.2. Verify that the analytic solution of the differential equation in 1.1 above is y = e**- and…

A: Given differential equation is y'=2xy Also, given that the analytic solution of the  differential…

Q: - It is known that y1 (x) = x and y2(x) = x are solutions of the differential equation xy" - 2xy +…

A:

Q: Prove that if f(x) solves a linear homogeneous differential equation with constant coef- ficients…

A:

Q: 10. Prove that if f(x) solves a linear homogeneous differential equation with constant coef-…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: For the differential equation (x2 – 9)y" + (x - 3)y' + 2y = 0, a regular singular point is: X-9 O…

A:

Q: Transform the following Euler differential equation into a constant coefficient linear differential…

A: From the given substitution z = ln(x)

Q: (6) Show that the differential equation (3ry + y*)dr + (r² + ry)dy = 0 is homoge- neous; solve it…

A:

Q: Verify that the indicated function y = (x) is an explicit solution of the given first-order…

A:

Q: Verify that the indicated function y = e(x) is an explicit solution of the given first-order…

A:

Q: Verify that the indicated function y = 4(x) is an explicit solution of the given first-order…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: (4x3 – 3y²)dx + 6xydy = 0 Ans: 2x3 + 3y² = 2x %3D

A: first will convert the given differential equation into linear differential equation of first order…

Q: 2. (a) Check that the first order differential equation 3x-3y = 10(xy") is homogeneous and hence…

A: Follow definition of homogenity of O.D.E Apply  substitution to solve given O.D.E

Q: 2. Obtain the general solution of x³y" + 3x2y' + x (+x) y = 0. Hint: The differential equation is…

A: Given: x3y''+3x2y'+x14+xy=0 To find: Bessel function.

Q: Let L(y) = x²y" – x(x+2)y' + (x+2)y %3D (a) consider the homogenous differential equation L(y) = 0…

A:

Q: 48. (a) Show that y(x) = Cx* defines a one-parameter fam- ily of differentiable solutions of the…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A: Note the following chain rule of differentiation : For a differentiable function y=1f(x)  , y' =…

Q: 2. Verify that the indicated expression is an implicit solution of the given differential equation.…

A: We try to obtain the differential equation from the given expression.

Q: 2. (a) Check that the first order differential equation 4x- dx dy - 4y =9(/xy² ) is homogeneous and…

A:

Q: 1. Verify that y-ciece of the nonhomogenec " as differential equation 5X + be is the general…

A:

Q: Verify that the indicated function y = P(x) is an explicit solution of the given first-order…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: Which of the following is the particular solution of a nonhomogeneous differential equation…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: 8. Suppose y1 and y2 are linearly independent, and the Wronskian of them, W (y1, Y2), is identically…

A:

Q: Verify that the indicated function y = Q(x) is an explicit solution of the given first-order…

A: Solution : when y=125-x2y'=-125-x22×-2xy'=2x25-x22y'=2xy2thus in terms of x,2xy2=dydx

Q: For the differential equation (x2 – 4)y" + (x – 2)y' + 2y = 0, a regular singular point is: x=2 O…

A:

Q: (х-1)у"-ху'+у-0 Yı(x)=e* y'2(t)=x a. verify if the functions y, and y2 are solution of the given…

A:

Q: 3. Consider the differential equation zy" - 6ry + 12y = 0. (a) Verify that and r form a fundamental…

A:

Q: We are given the following homogenous differential equation and pair of solutions on the given…

A: Given differential equation x2y''-7xy'+15y=0We have to verify that given function x3 and x5 form a…

Q: 3. Solve the ordinary differential equation 1 u'(x) 0. t-u?(x) with the initial condition u(2) = 0…

A: According to Runga kutta method  For an ordinary differential equation  dudt=f(t,u)…

Q: Consider the differential equation (E): x²y"-xy' +y 0. Given that y, x and y, xlnx are solutions of…

A: Given solutions are y1 = x  and  y2 = xln(x)

Q: Find the Particular solution of the differential equation : D2+3D+2y=3sin x , given that x=0, y=-0.9…

A:

Q: 6. Find the general solution to a fourth-order linear homogeneous differential equation for x(t)…

A: Given the two particular solutions 3e2t & 6t2e-t of a fourth-order linear homogeneous…

Q: Given that y, = 1, y2 = x2,y3 = e* are solutions of a third order differential equation (E). Then…

A:

Q: For the differential equation (x – 4)y" + (x– 2)y' + 2y = 0, the singular points are: O None of them…

A: Here we use the basic definitions of singular point

Q: Consider the differential equation (E): 2x²y" – xy' – 2y = 0. Given that y, = x2 is a solution of…

A:

Q: 11. Show that if n E Z the only solutions of the differential equation p² F" (r) + rF'(r) – n²F(r) =…

A: show that if n∈ℤ the only solution for the differential equation r2F''r+rF'r-n2Fr-0 which are twice…

Q: Given that y, (x) = x-1 and y, (x) = x3/2 are solutions of the differential equation 2x2y" + xy'-3y…

A:

Q: A student is solving a Cauchy-Euler differential equation and obtains the following solutions of the…

A:

Q: 2(a) Show that for any values of the arbitrary constant c, and the function C2 d²y + y = 0 y = c1sin…

A:

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

A:

Q: 8. Suppose y1 and y2 are linearly independent, and the Wronskian of them, W(y1, Y2), is identically…

A:

Q: Verify that the indicated function y g(x) is an explicit solution of the given first-order…

A:

Q: (a) Verify that the one-parameter family y-2y = x² -x + cis an implicit solution of the differential…

A: (a) The one parameter of family is given asy2−2y=x2−x+c___1diferentiating both the side wrt .x, we…

Q: Solve the ordinary differential equation 1 u'(x) – t - u?(x) with the initial condition u(2) = 0 by…

A: Given ordinary differential equation isu'(x)-1t-u2(x)=0Initial condition is u(2)=0The solution of…