Just as there are simultaneous algebraic equations (where a pair of numbers have
ID: 1948300 • Letter: J
Question
Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations. y1 = sin(x) + cos(x) y2 = cos(x) - sin(x) y1 = sin(x) y2 = cos(x) y1 = cos(x) y2 = -sin(x) y1 = e-x y2 = e-x y1 = ex y2 = ex y1 = e4x y2 = e4x y1 = 2e-2x y2 = 3e-2x As you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions.Explanation / Answer
Y1'=Y1-2Y2............Y2=0.5Y1-0.5Y1'.........................1
Y2'=3Y1-4Y2............................2
DIFFERENTIATING EQN.1 WE GET
Y2'=0.5Y1'-0.5Y1''..............................................3
SUBSTITUTING FOR Y2' FROM EQN.2 WE GET
3Y1-4Y2=0.5Y1'-0.5Y1''................4
SUBSTITUTING FOR Y2 FROM EQN.1 WE GET
3Y1-4[0.5Y1-0.5Y1']=0.5Y1'-0.5Y1''
3Y1-2Y1+2Y1'=0.5Y1'-0.5Y1''.......................5
MULTIPLYING WITH 2 AND SIMPLIFYING WE GET
Y1'' +3Y1' +2Y1=0..................................................6
SO WE GET
[D^2+3D+2]Y1=0
[(D+1)(D+2)]Y1=0
Y1=C1[E^(-T)]+C2[E^(-2T)]................................................7
Y1'=-C1[E^(-T)]-2C2[E^(-2T)]........................................8
PUTTING IN EQN.1
Y2=0.5[C1{E^(-T)+C2{E^(-2T)+C1{E^(-T)+2C2{E^(-2T)}]
Y2=C1[E^(-T)]+1.5C2[E^(-2T)]..............................................9
HENCE THE SOLUTION SET IS
Y1=C1[E^(-T)]+1.5C2[E^(-2T)]................................................7
Y2=C1[E^(-T)]+1.5C2[E^(-2T)].........OR.....MULTIPLYING WITH 2
Y2=2D1[E^(-T)]+3D2[E^(-2T)]...........................................9
[USED T WHILE YOUR ANSWER USED X FOR THE INDEPENDENT VARIABLE]
HENCE THE RIGHT CHOICE IS
OPTIONS ....D AND G
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.