Mltipl-Choic Tst Introdction to Partial Diffrntial Eqations COMPLETE SOLUTION SET 1. A partial diffrntial qation has (A on indpndnt variabl (B two or mor indpndnt variabls (C mor than on dpndnt variabl (D qal nmbr of dpndnt and indpndnt variabls Soltion Th corrct answr is (B. If a diffrntial qation has onl on indpndnt variabl thn it is calld ordinar diffrntial qation. A partial diffrntial qation has two or mor indpndnt variabls.
. A soltion to th partial diffrntial qation. is (A cos( (B + (C (D Soltion Th corrct answr is (D. W will solv this b sbstitting th givn choics. Th choic which satisfis th partial diffrntial qation is th corrct answr. Lt s start with option (A cos( ; cos( ; cos( Sbstitting in partial diffrntial qation cos( cos( Lt s start with option (B + ; ; Sbstitting in partial diffrntial qation Lt s start with option (C
cos( ; cos( ; Sbstitting in partial diffrntial qation Lt s start with option (D ; cos( ; Sbstitting in partial diffrntial qation
. Th partial diffrntial qation z z 5 + 6 is classifid as (A lliptic (B parabolic (C hprbolic (D non of th abov Soltion Th corrct answr is (A. A gnral scond ordr partial diffrntial qation with two indpndnt variabls is of th form A + B + C + D 0 whr A, B, andc ar fnctions of and and D is a fnction of,, and,. Th abov PDE can b rwrittn as z z z 5 + 0 + 6 0 Dpnding on th val of B 4AC, th nd ordr linar PDE can b classifid into thr catgoris. 1. if B 4AC < 0, it is calld lliptic. if B 4AC 0, it is calld parabolic. if B 4AC > 0, it is calld hprbolic In th abov qstion, A 5, B 0, C 6, giving B 4AC 0 4(5(6 10 < 0 This classifis th diffrntial qation as lliptic.
4. Th partial diffrntial qation z z 5 is classifid as (A lliptic (B parabolic (C hprbolic (D non of th abov Soltion Th corrct answr is (B. A gnral scond ordr partial diffrntial qation with two indpndnt variabls is of th form A + B + C + D 0 whr A, B, andc ar fnctions of and and D is a fnction of,, and,. Th abov PDE can b rwrittn as z z z z 0 + 0 + 5 0 Dpnding on th val of B 4AC, th nd ordr linar PDE can b classifid into thr catgoris. 1. if B 4AC < 0, it is calld lliptic. if B 4AC 0, it is calld parabolic. if B 4AC > 0, it is calld hprbolic In th abov qstion, A 0, B 0, C 5, giving B 4AC 0 4(0(5 0 This classifis th diffrntial qation as parabolic.
5. Th partial diffrntial qation z z 5 0 is classifid as (A lliptic (B parabolic (C hprbolic (D non of th abov Soltion Th corrct answr is (C. A gnral scond ordr partial diffrntial qation with two indpndnt variabls is of th form A + B + C + D 0 whr A, B, andc ar fnctions of and and D is a fnction of,, and,. Th abov PDE can b rwrittn as z z z 1 + 0 5 0 Dpnding on th val of B 4AC, th nd ordr linar PDE can b classifid into thr catgoris. 1. if B 4AC < 0, it is calld lliptic. if B 4AC 0, it is calld parabolic. if B 4AC > 0, it is calld hprbolic In th abov qstion, A 1, B 0, C 5, giving B 4AC 0 4(1( 5 0 > 0 This classifis th diffrntial qation as hprbolic.
6. Th following is tr for th following partial diffrntial qation sd in nonlinar mchanics known as th Kortwg-d Vris qation. w w w + 6w 0 t (A linar; rd ordr (A nonlinar; rd ordr (B linar; 1 st ordr (C nonlinar; 1 st ordr Soltion Th corrct answr is (B. Th partial diffrntial qation is nonlinar bcas th cofficint of th drivativ trm a fnction of th dpndnt variabl, w. Th qation is a rd ordr as that is th highst drivativ in th partial diffrntial qation. w is