In each of Problems 10 through 15, verify that the given functions y₁ and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneousequation. In Problems 14 and 15, g is an arbitrary continuous function.11. t²y" -t(t+2)y' + (t+2)y= 2t³, t>0; y₁(t)=t, y₂(t) = tet

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Answered: In each of Problems 10 through 15,…

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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This is a linear, first-order, non-homogeneous equation of the form y'+p(x)y=f(x), so you will use variation of parameters to solve it. Remember that there are two steps: Find any solution of the associated homogenous equation, y1.Set y= uy1 where u is a mystery function; you can use the formula that u'=f(x)/y1. Which should Find the general solution for the following differential equation
SI have a slope of zero. SMy y-intercept is positive. SMy solutions are above the line. S(0,4) is a solution. O M. O N. O L O G O H.
If the length of a curve from (0,–3) to (3,3) is given by [ V1+(x² – 1)² dx , which of the following could be an equation for this curve? x (A) у%3 3 - 3 (В) у 3D -- 3x – 3 3 (С) у%3D—— х-3 3 (D) y =-+x- 3 3
Solve B16b. and perform four iterations.
II. Solve the following Bernoulli's equations. 1. y'(t) = 3y – y. Answer. y =esy and y = 0. 1+ce- 2. y' -ty = y, y(2) = -2. Answer. y =x 2-x2 3. xy' + y + xy = 0, y(1) = 2. Answer. y = x(1+2 In x) 4. y' +y = xy', y(0) = -1. Answer. y = - 5. 2 = dy +2x dx Answer. y = +V-1 – 2x + ce2x, 6. y' + xV = 3y. Answer, y = 2 (; + + c*}, and y = 0. Hint. When dividing the equation by Vy. one needs to check if y = 0 is a solution, and indeed it is. 7. y' + y = -xy². Answer. y = " and y = 0. ce-x-1 8. y' + xy = y", y(1) = -! Answer. y = - 1.7. Numerical Solution by Euler's Method 35 9. The equation dy y? + 2x xp y could not be solved in the preceding problem set because it is not homoge- neous. Can you solve it now? Answer. y =tVcezx – 2x – 1. 10. y' = ex +y. Answer. y = te/x² + c. . 11. Solve the Gompertz population model (a and b are positive constants) dx dt = x (a – b ln x), x> 1. Hint. Setting y = In x, obtain y = a – by. Answer. x(t) = q@/bce-br 12. Solve x(y - e) + 2 = 0. Hint. Divide the…
Find a general solution. 2y'' + 7y' - 4y=0 y(t) =
Solve problem 5 with explanation please. Thank you
2. Determine the solution of each the following equations which satisfies the given conditions: a. y" +5y" + 4y = 0, in %3D y = y"= 0 when x= 0 & y = 1, y' 2 when x= n/2 b. y"-3y" + 4y = 0, %3D y = 1, y'= -8, y"= -4 when x 0 %3D %3D c. y" +y"-0, y(0) = 2, y(0) = 1, y"(0) = -1 = -] %3D
. Solve the following problem (a) Y'(t) = XY (t) + 1, (b) Y'(t) = \Y(t)+t, Y(0) = 1. Y(0) = 3.
10. The acceleration of a particle is given by a(t) = 6t – 14 + 9e¯t, and the initial velocity is v(0) = 3 m/s, and the intitial position is p(0) = 4 m. Determine its position function p(t).
This is a linear, first-order, non-homogeneous equation of the form y'+p(x)y=f(x), so you will use variation of parameters to solve it. Remember that there are two steps: Find any solution of the associated homogenous equation, y1.Set y= uy1 where u is a mystery function; you can use the formula that u'=f(x)/y1. Which should Find the general solution for the following differential equation
Hh.101.

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