Consider a particular solution $ y_p = y_1 v_1 + y_2 v_2 + y_3 v_3 $ for the following Cauchy-Euler equation:\[ x^3 y''' + x^2 y'' - 2x y' + 2y = x^2 \]where $ y_1 = x $, $ y_2 = x^{-1} $, $ y_3 = x^2 $ are linearly independent solutions corresponding to the homogeneous equation. Find the general solution.

Let x = eu ⇒u = ln(x). and xD = r x2D2=rr-1x3D3=rr-1r-2Substituting in x3y'''+x2y''-2xy'+2y=x2x3D3+x2D2-2xD+2y=x2rr-1r-2+rr-1-2ry=e2uHomogeneous equation,rr-1r-2+rr-1-2r+2y=0Characteristic equation rr2-2r-1+2=0⇒r=1,-1,2.Complementary Function yc= c1y1+c2y2+c3y3= c1x+c2x-1+c3x2Particular solution is yp=1D3-2D2-D+2e2u=1D+13-2D+12-D+1+2e2u=1DD2+D-2e2u=x23ln(x-49 General solution y= yc + yp = c1x+c2x-1+c3x2+x23ln(x-49

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Advanced Math

Consider a particular solution Yp = Y1V1 + Y2V2 + Y3V3 for the following Cauchy-Euler equation x*y" + x?y" – 2ry' + 2y = x² - where y = x, Y2 = x-1, y3 = x² are linearly independent solution corresponding to homogeneous equation. Find the general solution.

Consider a particular solution Yp = Y1V1 + Y2V2 + Y3V3 for the following Cauchy-Euler equation x*y" + x?y" – 2ry' + 2y = x² - where y = x, Y2 = x-1, y3 = x² are linearly independent solution corresponding to homogeneous equation. Find the general solution.

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

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Consider a particular solution $ y_p = y_1 v_1 + y_2 v_2 + y_3 v_3 $ for the following Cauchy-Euler equation:

\[ x^3 y''' + x^2 y'' - 2x y' + 2y = x^2 \]

where $ y_1 = x $, $ y_2 = x^{-1} $, $ y_3 = x^2 $ are linearly independent solutions corresponding to the homogeneous equation. Find the general solution.

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