MHB Finding General Solution vs Solving Differential Equations

[ifeg]

If you are asked to solve a differential equation (single order) that has no initial values, then you separate if possible, integrate and include the Constant of integration, leaving the response in terms of the dependent variable, right?

So if you are given a single order differential equation, and asked to find the general solution, what do you do differently?

eg: $$ \frac {dy}{dx} - y = e^{3x} $$

 

[CaptainBlack]

If you are asked to solve a differential equation (single order) that has no initial values, then you separate if possible, integrate and include the Constant of integration, leaving the response in terms of the dependent variable, right?

So if you are given a single order differential equation, and asked to find the general solution, what do you do differently?

eg: $$ \frac {dy}{dx} - y = e^{3x} $$

Nothing.

CB

 

[poissonspot]

It depends on what you mean by general solution.
$\frac {dy}{dx} - y = e^{3x}$
In this case you have a linear differential equation.
If you wanted the general solution for a linear differential equation then you need to recognize the general form $\frac {dy}{dx} + P(x)y=Q(x)$ , and work with that.
If you are asking about a particular diff. eq then the general solution is nothing more than what you have described.

 

[ifeg]

No, i don't think there's anything about a particular diff equation. i came across a question that asked just to find the general solution, but it had come after some other questions that asked to solve some diff equations, so i was wondering what, if any, was the difference between the two. I did some research and did see info about what you had suggested, with the P(x) y = Q(x) so i was wondering if that applied. Thanks for your assistance.

It depends on what you mean by general solution.
$\frac {dy}{dx} - y = e^{3x}$
In this case you have a linear differential equation.
If you wanted the general solution for a linear differential equation then you need to recognize the general form $\frac {dy}{dx} + P(x)y=Q(x)$ , and work with that.
If you are asking about a particular diff. eq then the general solution is nothing more than what you have described.

 

[Telemachus]

I think that general solution should reefer to the one you got without the initial values, and the a singular solution would be the one that you obtain after replacing the initial values.

 

[Prove It]

The general solution is the FAMILY of functions that satisfies the Differential Equation (i.e. has the arbitrary constants).

The particular solution is the single solution of the Differential Equation that satisfies BOTH the DE AND the initial/boundary conditions.

 

[Cadbury]

If you are asked to solve a differential equation (single order) that has no initial values, then you separate if possible, integrate and include the Constant of integration, leaving the response in terms of the dependent variable, right?

So if you are given a single order differential equation, and asked to find the general solution, what do you do differently?

eg: $$ \frac {dy}{dx} - y = e^{3x} $$

Hi! When you are given conditions in the problem, for example y(0)=1 and y'(0)=0, you substitute this to the general solution which refers to the solution you have obtained from seperating and differentiating and you will obtain your partial solution :)