20. Consider the third-order linear homogeneous ODE Arty" + Boy" + Cay' + Dy = 0, where A, B. ( and D are constants. Make the substitution + = e" into the latter ODE, and write down the resulting ODE inside the box provided below.& . Given third order linear homogeneous ODE A x3y'' + Buzy" + (xy' + Dy = 0 - A where A, B. C, D are constants. To make the substitution n= el We fiend have to find the derivative of of with respect to a n in terms of U. using chain sule u u = e taking log both side + log n = u loge u = loge u du du = -1 duz 2 2 2 = ) 2 3 3 Now dy dy x du 4 du du du du dy ( dy dy X 2 du du 4 11 dzy - dy * x 5 2 2 dudy -le dzy du 3 du Ju x ( x ) - 2 8 x ( + ) 2 3 dy dus + ) 3 day duz x ( * ) + 2 dy New substituting x = all in the original ODE, and derivatives which are solved above in the ean (A) using ean (4, 6) and 6 3 45 = ) A e d'y dus ( ) - 3 dy duz ( zu ) + 2 dy ( give) + B e d'y x dy 24 e du e - u + ce u du + Dy = 0 > A day 3 Ady + 2 Ady + Body Body du ? duz du duz du + c dy du + Dy = 0 = ) A d' y + ( - 3 A + B ) dy + ( 2 A - B + c ) dy + Dy = 0 dus duz du which is final tuswer