- #1
artan
- 7
- 0
Please if anyone can help me to solve this differential equation.
Please if anyone can help me to solve this differential equation.
- Thread starter artan
- Start date
In summary, the conversation discusses solving a differential equation and finding the third solution, which is a special function. The general solution to the ODE is also provided, along with a link to the special function called exponential integral. The conversation also mentions a method for solving the equation and asks for further clarification and steps. The conversation ends with a request for help and a thank you.
- #2
JJacquelin
- 801
- 35
Hello,
Two obvious solution of the homogeneous part of the equation are y=x and y=x². The third is a special function.
Nevertheless, particular solutions for the whole ODE can be derived :
Let y=x*f(x) and solve the ODE which unknown is f(x).
- #3
artan
- 7
- 0
I can find that first and second obvious solution of the homogeneous part x and x^2,but how can I find the third,it is something of x^x.
Can anyone explain how to find the particular solution showing me some steps of the solution.Thank you
- #4
artan
- 7
- 0
Can you write the special function and show me some steps how to find particular solution.Thank you
- #5
kosovtsov
- 70
- 0
The general solution to your ODE is as follows
[itex]y(x) = -3x(x-3)\ln(x)+[(-\frac{1}{2}x^2+x)\int_{-x}^∞ \frac{\exp(-t)}{t}dt -\frac{1}{2}\exp(x)(x-1)]C_1-\frac{x^3+9}{2}+x^2C_2+xC_3[/itex]
where [itex]C_i[/itex] are arbitrary constants.
- #6
JJacquelin
- 801
- 35
The special function is Ei(X)
http://mathworld.wolfram.com/ExponentialIntegral.html
As allready said, let y(x)=x*f(x) which leads to
x f '''+(2-x)f ''= x +(3/x²)
Ei(-x) will appear in solving this equation.
- #7
artan
- 7
- 0
I started solving this DE this way:
y=x^m
y'=m*x^(m-1)
y''=m(m-1)x^(m-2)
y'''=m(m-1)(m-2)x^(m-3)
and replace them in DE we get:
(m-1)(m-2)(m*x^(m-1)-x^m)=0
so we get :
y1=x
y2=x^2
m*x^(m-1)-x^m=0 this is the third solution but i don't know how to find it
thank you.
- #8
JJacquelin
- 801
- 35
artan said:
You suppose that the third solution is on the patern y=x^m which is not the case. As a consequence this method cannot lead to the third solution.
By chance, the first and the second solution are on the patern y=x^m, so leading to m=1 and m=2. But obviously not the third.
- #9
artan
- 7
- 0
MR.Kosovtsov can you write the whole method how did you get the solution,not just the final solution.
I will be grateful.
Thank you.
Last edited:
- #10
artan
- 7
- 0
which method should i use to get the solution.
Can you write for me the beginig of the method you use?
I will be grateful.
Thank you for the help.
Last edited:
- #11
- #12
artan
- 7
- 0
Thank you for the help
FAQ: Please if anyone can help me to solve this differential equation.
How do I know if a differential equation is solvable?
Not all differential equations have analytical solutions, meaning they cannot be solved using traditional mathematical methods. In general, you can determine if an equation is solvable by checking its order and linearity. First, the order of the equation refers to the highest derivative present. If the order is greater than 2, it may be more difficult to solve. Additionally, a linear equation has terms with derivatives that are not multiplied or divided by each other. If the equation is not linear, it may be more challenging to find a solution.
What are some common methods for solving differential equations?
Some common methods for solving differential equations include separation of variables, integrating factors, and substitution. Separation of variables involves isolating the dependent and independent variables on opposite sides of the equation and integrating each side separately. Integrating factors involve multiplying both sides of the equation by a specific function to make the equation easier to solve. Substitution involves replacing the dependent variable with a new variable to simplify the equation.
Can I use a computer to solve differential equations?
Yes, there are many software programs and coding languages that can be used to solve differential equations. These programs use numerical methods to approximate solutions, which can be helpful for more complex equations that do not have analytical solutions. However, it is still important to have a basic understanding of differential equations and their solutions in order to use these programs effectively.
What are initial conditions in a differential equation?
Initial conditions refer to the values of the dependent variable and its derivatives at a specific point in the domain of the equation. These conditions are typically given as part of the problem and are used to find a particular solution to the equation. In some cases, the initial conditions may be used to find a general solution, which includes a constant term, that can then be solved for using additional information.
Are there real-world applications for differential equations?
Yes, differential equations are used to model many real-world situations in fields such as physics, engineering, biology, and economics. For example, they can be used to model the motion of a swinging pendulum, the rate of population growth, or the flow of electricity in a circuit. Solving these equations allows scientists and engineers to make predictions and understand the behavior of complex systems.