Question

Solve the Cauchy Euler DE 2%Y" + 51Y - 21y = 0. List the two values that you get for m. List the smallest first and then the larger.

Answer 1

0 x²y" + say'-21y=0 Quest So Euler DE. List the two valued that you get for m Hist the smalles first and then the larger. An Euler ODE has the for on assume of the Given Dit is, x²y" + 5x4'-214:0 A second order cauchy. homogeneous has the general form axlyrt baytey- ax²y + bay't cy=0 solution form ar (called Euler's substitution). so, Rewrite the equation with y = xh. since it is solution, hence it will satisfy the egr 0 A So, yragya rir-la-2 Now O becomes rer-1)x² a +5ruari - 21% = 0 3) x Flr-1) +5v-2] (A.E.) [r² +41-21] = 2 [ +'+75 - 3r-21] =- get (r+7) (6-3) = 21 r=07 and r=3 so for two real roots rith the generel solution of equation & takes the form Y = C, XtGxl, r.2 20 ... Auxialy eqm) ar 7 arto 다.

Here, solt is y - Cpx' + x2x7 X₂ x 7. Ans. which means the two values of m (which is noth thing but are 7 and 3. Hence, Am r ) m = -7 m₂ = 3