Some questions regarding a differential equation

  • #1

Main Question or Discussion Point

Hello,

I am new here. I hope I am posting my problem at the right place.

I need some urgent help regarding the following differential equation:

A[itex]\frac{d^{2}y}{dx^{2}}[/itex]+B[itex]\frac{dy}{dx}[/itex]=f(x,[itex]\lambda[/itex]).......(1)

where, A and B are constants. x and [itex]\lambda[/itex] are independent.

I have solved the equation analytically and the solution is a function of x and [itex]\lambda[/itex], i.e y(x,[itex]\lambda[/itex]).

Afterwards, to obtain y(x) [i.e to make it [itex]\lambda[/itex] independent], I have integrated the solution over a region of [itex]\lambda[/itex], i.e [itex]\int{y(x,\lambda)}[/itex]d[itex]\lambda[/itex]=y(x) ........(2)

My question is - can I integrate the term f(x,[itex]\lambda[/itex]) over a region of [itex]\lambda[/itex] before solving the eqn (1)? i.e if I replace f(x,[itex]\lambda[/itex]) by [itex]\int{f(x,\lambda)}[/itex]d[itex]\lambda[/itex] = f(x) in eqn (1) and solve the following equation:

A[itex]\frac{d^{2}y}{dx^{2}}[/itex]+B[itex]\frac{dy}{dx}[/itex]=f(x) ............(3)

Is it a correct way to solve (1)?

I have solved the eqn (3). But, the solution (y(x) curve) is completely different than (2).

Which method should I use (first or second method)? Any explanation of getting different results would also help me. Please pardon my ignorance - I am new in this field.

Many many thanks in advance.........
 

Answers and Replies

  • #2
346
48
Yes if x and lambda are independent, then it does not matter if you integrate before or after you solve the ode.
 

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