# Find a particular solution that satisfies the intial condition

• Ki-nana18
In summary, a particular solution is a specific solution to a differential equation that satisfies the given initial conditions. To find a particular solution, the differential equation must first be solved using appropriate methods and then the initial conditions can be used to determine the specific values of the constants in the general solution. This is important because it allows us to determine the exact solution that satisfies the initial conditions, providing valuable insights in various fields. If a particular solution cannot be found, it may be necessary to re-evaluate the problem or consider alternative methods. A particular solution can also be represented graphically by plotting the equation on a graph.
Ki-nana18

## Homework Statement

2xy'-ln x2=0 y(1)=2

## The Attempt at a Solution

2x(dy/dx)-ln x2=0

I think I'm suppose to separate variables and then integrate next but I'm not sure.

Yes, this is a separable differential equation.

You should use the fact that ln(x2) = 2ln(x).

2x*dy/dx = 2 ln(x)
$$\Rightarrow~dy~=~ \frac{ln(x)}{x}dx$$

Now integrate both sides. Don't forget the constant of integration, which you will determine from your initial condition, y(1) = 2.

## 1. What is a particular solution?

A particular solution is a specific solution to a differential equation that satisfies the given initial conditions. It is often denoted as yp or simply as y.

## 2. How do you find a particular solution?

To find a particular solution, you must first solve the differential equation using appropriate methods such as separation of variables or substitution. Then, you can plug in the given initial conditions to determine the specific values of the constants in the general solution, resulting in the particular solution.

## 3. Why is finding a particular solution important?

Finding a particular solution allows us to determine the exact solution to a differential equation that satisfies the initial conditions. This can provide valuable information and insights in various fields such as physics, engineering, and economics.

## 4. What happens if a particular solution cannot be found?

If a particular solution cannot be found, it is possible that the given initial conditions are not compatible with the given differential equation. In this case, it may be necessary to re-evaluate the problem or consider alternative methods of solving the equation.

## 5. Can a particular solution be represented graphically?

Yes, a particular solution can be represented graphically by plotting the equation on a graph with the independent variable on the x-axis and the dependent variable on the y-axis. The graph will show the specific behavior of the solution that satisfies the initial conditions.

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