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Conceptual Second order differential eqn question

by CAF123
Tags: conceptual, differential, order
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CAF123
#1
Nov22-12, 09:49 AM
PF Gold
P: 2,314
I know that if ##Y_1## and ##Y_2## are two solutions of a nonhomogeneous second order differential eqn, then ##Y_1 - Y_2## is also a solution. So this motivates the following: if we set ##Y_1 = y(x)##, where ##y(x) ## is an arbritary soln of the nonhomogeneous ODE and ##Y_2 = y_p(x)##, some particular soln, we get that ##Y_1 - Y_2## is a solution to the corresponding complementary equation, $$ a(Y_1 - Y_2)'' + b(Y_1 - Y_2)' + c(Y_1 - Y_2) = 0,$$ ie ##y_c(x) = Y_1 - Y_2 = y(x) - y_p(x)##.
I have two questions:
1)Why set ##Y_1 = y(x)## and ##Y_2 = y_p(x)##? Could we have set ##Y_1 = y_p(x) ##and ##Y_2 = y(x)## so that in the end we get ##y(x) = y_p(x) - y_c(x),## where to recover the usual ##y(x)= y_p(x) +y_c(x),## we introduce an arbritary negative for the constants in the ##y_c(x)## term?

2) I often read questions: Find the general soln of ... and given the initial conditions...find the particular soln. I can do these questions fine. Conceptually though and understanding what is going on, I get a little confused here because we have already defined the term 'particular soln' (as above ##y_p(x)##) in order to find the general soln. So is this two different things with the same term attached to them? I recall that ##y_p(x)## is sometimes called the particular integral?
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HallsofIvy
#2
Nov22-12, 11:27 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
Quote Quote by CAF123 View Post
I know that if ##Y_1## and ##Y_2## are two solutions of a nonhomogeneous second order differential eqn, then ##Y_1 - Y_2## is also a solution.
NO! You don't know that- it is not true. If [itex]Y_1[/itex] and [itex]Y_2[/itex] are two solutions of the same nonhomogeneous differential equation, then [itex]Y_1- Y_2[/itex] is a solution to the associated non-homogeneous equation.

So this motivates the following: if we set ##Y_1 = y(x)##, where ##y(x) ## is an arbritary soln of the nonhomogeneous ODE and ##Y_2 = y_p(x)##, some particular soln, we get that ##Y_1 - Y_2## is a solution to the corresponding complementary equation, $$ a(Y_1 - Y_2)'' + b(Y_1 - Y_2)' + c(Y_1 - Y_2) = 0,$$ ie ##y_c(x) = Y_1 - Y_2 = y(x) - y_p(x)##.
Okay, now that is true- and is not what you said above.

I have two questions:
1)Why set ##Y_1 = y(x)## and ##Y_2 = y_p(x)##? Could we have set ##Y_1 = y_p(x) ##and ##Y_2 = y(x)## so that in the end we get ##y(x) = y_p(x) - y_c(x),## where to recover the usual ##y(x)= y_p(x) +y_c(x),## we introduce an arbritary negative for the constants in the ##y_c(x)## term?
I don't see any difference. You are just changing which function you call [itex]Y_1[/itex] and which you call [itex]Y_2[/itex].

2) I often read questions: Find the general soln of ... and given the initial conditions...find the particular soln. I can do these questions fine. Conceptually though and understanding what is going on, I get a little confused here because we have already defined the term 'particular soln' (as above ##y_p(x)##) in order to find the general soln. So is this two different things with the same term attached to them? I recall that ##y_p(x)## is sometimes called the particular integral?
Yes, "particular integral" is a better term than "particular solution".
CAF123
#3
Nov22-12, 12:09 PM
PF Gold
P: 2,314
Quote Quote by HallsofIvy View Post
NO! You don't know that- it is not true. If [itex]Y_1[/itex] and [itex]Y_2[/itex] are two solutions of the same nonhomogeneous differential equation, then [itex]Y_1- Y_2[/itex] is a solution to the associated non-homogeneous equation.
Should that be the 'associated homogeneous' eqn?


I don't see any difference. You are just changing which function you call [itex]Y_1[/itex] and which you call [itex]Y_2[/itex].
Changing the order will mean that the general solution is the complementary function - the particular integral rather than the complementary function + the particular integral, no?(since it is strictly ##Y_1 - Y_2##)

Yes, "particular integral" is a better term than "particular solution".
Ok, thanks. So ##y_p(x)## should really be called particular integral so as to avoid confusion. Where did the 'integral' come from in its name?

CAF123
#4
Nov24-12, 11:55 AM
PF Gold
P: 2,314
Conceptual Second order differential eqn question

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