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X
- LET Y OF T REPRESENT YOUR BANK ACCOUNT BALANCE IN DOLLARS
AFTER 2 YEARS.
SUPPOSE YOU START WITH $60,000 IN THE ACCOUNT.
EACH YEAR THE ACCOUNT EARNS 3% INTEREST
AND YOU WITHDRAW $4,000.
WE'RE GOING TO WRITE THE DIFFERENTIAL EQUATION
MODELING THIS SITUATION.
SO THE FIRST THING TO RECOGNIZE HERE
IS THAT Y OF T WOULD BE THE ACCOUNT BALANCE IN DOLLARS
AND THEREFORE Y PRIME OF T WHICH WE COULD ALSO WRITE AS DYDT
WOULD BE THE CHANGE IN Y WITH RESPECT TO T
OR THE CHANGE IN THE ACCOUNT BALANCE IN DOLLARS
WITH RESPECT TO TIME.
AND THERE ARE TWO THINGS HERE THAT AFFECT THE ACCOUNT BALANCE.
THE ACCOUNT BALANCE INCREASES BY ENTERING 3% INTEREST
BUT THEN IT ALSO DECREASES
BECAUSE YOU WITHDRAW $4,000 EACH YEAR.
TO DETERMINE THE INCREASE FROM THE INTEREST EARNED
WE WOULD CONVERT 3% TO A DECIMAL
AND MULTIPLY IT BY THE CURRENT ACCOUNT BALANCE
WHICH SHOULD BE Y OF T OR Y.
SO WE CAN SAY DYDT = 0.03 x Y.
BUT THEN BECAUSE YOU ALSO WITHDRAW $4,000 PER YEAR
WE'D HAVE TO SUBTRACT 4,000 AS WELL.
SO DYDT = 0.03Y = 4,000.
THIS WOULD MEASURE THE CHANGE IN THE ACCOUNT BALANCE
PER UNIT OF TIME
WHERE TIME IS IN YEARS.
WE'RE ALSO ASKED TO FIND Y OF 0.
NOTICE HOW THIS IS Y OF 0 NOT Y PRIME OF 0
WHERE AGAIN THE FUNCTION Y GIVES US THE ACCOUNT BALANCE.
SO Y OF 0 REPRESENTS THE STARTING ACCOUNT BALANCE
WHICH WE'RE TOLD IS $60,000
AND THEREFORE Y OF 0,
THE STARTING ACCOUNT BALANCE, IS 60,000.
I HOPE YOU FOUND THIS HELPFUL.