Conceptual Second order differential eqn question

In summary, the conversation discusses the relationship between two solutions of a nonhomogeneous second order differential equation and the corresponding complementary equation. It is shown that setting ##Y_1 = y(x)## and ##Y_2 = y_p(x)##, where ##y(x)## is an arbitrary solution of the nonhomogeneous ODE and ##y_p(x)## is a particular solution, leads to ##Y_1 - Y_2## being a solution to the complementary equation. It is noted that switching the order of the functions does not change the result. The term "particular solution" is seen as potentially confusing and it is suggested that "particular integral" may be a better term. The origin of the
  • #1
CAF123
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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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  • #2
CAF123 said:
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".
 
  • #3
HallsofIvy said:
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?
 
  • #4
Can you help?
 
  • #5


1) The reason we set ##Y_1 = y(x)## and ##Y_2 = y_p(x)## is because we want to use the fact that the difference between two solutions of a nonhomogeneous second order differential equation is also a solution. By setting ##Y_1 = y(x)##, we are essentially creating a placeholder for any arbitrary solution to the nonhomogeneous equation. Similarly, by setting ##Y_2 = y_p(x)##, we are creating a placeholder for a particular solution to the nonhomogeneous equation, which we can find by using a method such as variation of parameters or the method of undetermined coefficients. If we had set ##Y_1 = y_p(x)## and ##Y_2 = y(x)##, then the difference ##Y_1 - Y_2## would not necessarily be a solution to the complementary equation. It is important to note that the order in which we set ##Y_1## and ##Y_2## does not matter, as long as we are consistent in our use of these placeholders.

2) The term "particular solution" can have different meanings depending on the context. In the context of finding the general solution to a nonhomogeneous equation, the particular solution refers to a specific solution that satisfies the nonhomogeneous equation, and is used to find the general solution. In this case, it is also sometimes referred to as the particular integral. However, in other contexts, the particular solution may refer to a solution that satisfies certain initial conditions or boundary conditions. It is important to carefully read the question or context to understand what is being asked for when the term "particular solution" is used.
 

1. What is a conceptual second order differential equation?

A conceptual second order differential equation is a mathematical equation that describes the relationship between the second derivative of a function and the function itself. It represents the acceleration of a system or phenomenon over time.

2. How is a second order differential equation different from a first order differential equation?

A second order differential equation involves the second derivative of a function, while a first order differential equation only involves the first derivative. This means that a second order equation takes into account the acceleration of a system, while a first order equation only takes into account its velocity.

3. What are some real-world applications of second order differential equations?

Second order differential equations are commonly used in physics, engineering, and other scientific fields to model and understand systems that involve acceleration and change over time. Examples include oscillating systems, electrical circuits, and motion of objects under the influence of gravity.

4. How do I solve a second order differential equation?

The process of solving a second order differential equation involves finding a general solution and then applying initial conditions to find a particular solution. This can be done using various methods such as separation of variables, substitution, or using a power series. It is important to have a strong understanding of calculus and algebra to successfully solve second order differential equations.

5. Can a second order differential equation have multiple solutions?

Yes, a second order differential equation can have multiple solutions. This is because there are typically two arbitrary constants in the general solution, which can result in an infinite number of possible solutions when initial conditions are applied. In some cases, a second order differential equation may also have no solution or a single unique solution.

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