Finding General Solution vs Solving Differential Equations

Click For Summary
SUMMARY

The discussion focuses on the distinction between finding the general solution and solving a particular differential equation, specifically a first-order linear differential equation represented by the form $$\frac{dy}{dx} + P(x)y = Q(x)$$. Participants clarify that the general solution includes arbitrary constants and does not require initial values, while the particular solution satisfies both the differential equation and specific initial conditions. An example provided is $$\frac{dy}{dx} - y = e^{3x}$$, illustrating the process of separating variables and integrating to find the general solution.

PREREQUISITES
  • Understanding of first-order linear differential equations
  • Familiarity with the method of separation of variables
  • Knowledge of integration techniques
  • Concept of initial and boundary conditions in differential equations
NEXT STEPS
  • Study the general form of linear differential equations: $$\frac{dy}{dx} + P(x)y = Q(x)$$
  • Learn the method of integrating factors for solving linear differential equations
  • Explore the concept of particular solutions and how they relate to initial conditions
  • Practice solving various first-order differential equations with and without initial values
USEFUL FOR

Students, educators, and professionals in mathematics or engineering who are looking to deepen their understanding of differential equations and their solutions.

ifeg
Messages
8
Reaction score
0
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: $$ \frac {dy}{dx} - y = e^{3x} $$
 
Physics news on Phys.org
ifeg said:
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
 
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.
 
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.
conscipost said:
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.
 
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.
 
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.
 
ifeg said:
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 :)
 

Similar threads

Undergrad General solution vs particular solution
  • · Replies 52 ·
2
Replies
52
Views
8K
Undergrad Question about the general solution to Hooke's law
  • · Replies 3 ·
Replies
3
Views
2K
MHB Help with a question (Bernoulli General solution)
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Number of solutions to the order of the equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
1K
MHB General Solution of Differential Equation System
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Find the general solution for the differential equation
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Partial Differential Equation solved using Products
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Solution to a differential equation with variable coefficients
  • · Replies 6 ·
Replies
6
Views
2K