# Partial differential equations problem - finding the general solution

• MHB
• Another1
In summary, the conversation discussed the general solution for the given equation 4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u , u(x,0)=4e^{-x}-e^{-5x} and how to solve it using the method of separation of variables. It was shown that the general solution can be expressed as a sum of solutions for different values of k.
Another1
$$\displaystyle 4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u$$ , $$\displaystyle u(x,0)=4e^{-x}-e^{-5x}$$

let $$\displaystyle U =X(x)T(t)$$

so

$$\displaystyle 4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT$$
$$\displaystyle 4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3$$
$$\displaystyle \left( 4\frac{\partial T}{T \partial t}-3 \right) +\frac{\partial X}{X \partial x} = 0$$

let K = constant
$$\displaystyle \frac{\partial X}{X \partial x} =\left( 3 - 4\frac{\partial T}{T \partial t} \right) = k$$
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$$\displaystyle \frac{\partial X}{X \partial x} = k$$
$$\displaystyle \frac{d X}{X} = k dx$$
$$\displaystyle X = C_1e^{kx}$$
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$$\displaystyle \left( 3 - 4\frac{\partial T}{T \partial t} \right) = k$$
$$\displaystyle 4\frac{\partial T}{T \partial t}= 3 - k$$
$$\displaystyle \frac{\partial T}{T \partial t}= \frac{1}{4}(3 - k)$$
$$\displaystyle \frac{d T}{T}= \frac{1}{4}(3 - k) dt$$
$$\displaystyle T = C_2e^{\frac{1}{4}(3 - k) t}$$
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general solution
$$\displaystyle u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t}$$ then $$\displaystyle C=C_1C_2$$

$$\displaystyle u(x,0) = C e^{kx} = 4e^{-x}-e^{-5x}$$ <<How do I solve this equation?

Another said:
general solution
$$\displaystyle u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t}$$ then $$\displaystyle C=C_1C_2$$

$$\displaystyle u(x,0) = C e^{kx} = 4e^{-x}-e^{-5x}$$ <<How do I solve this equation?
You have shown that $$\displaystyle u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t}$$ is a solution. But it is not the general solution. Instead, it provides a solution for every value of $k$. So a more general solution would be given by a sum of those solutions, for different values of $k$: $u(x,t) = \sum_k C_k e^{kx}e^{\frac{1}{4}(3 - k) t}.$ In particular, you could take the solutions for $k=-1$ and $k=-5$, and use their sum as a solution.

Opalg said:
You have shown that $$\displaystyle u(x,t) = C e^{kx}e^{\frac{1}{4}(3 - k) t}$$ is a solution. But it is not the general solution. Instead, it provides a solution for every value of $k$. So a more general solution would be given by a sum of those solutions, for different values of $k$: $u(x,t) = \sum_k C_k e^{kx}e^{\frac{1}{4}(3 - k) t}.$ In particular, you could take the solutions for $k=-1$ and $k=-5$, and use their sum as a solution.

Great! thank you

## 1. What is a partial differential equation?

A partial differential equation (PDE) is a mathematical equation that involves partial derivatives of an unknown function with respect to two or more independent variables. It is commonly used to describe physical phenomena in fields such as physics, engineering, and economics.

## 2. What is the general solution to a partial differential equation problem?

The general solution to a PDE problem is a solution that satisfies the equation for all possible values of the independent variables. It is a mathematical expression that contains arbitrary constants and can be used to find specific solutions for a given set of initial or boundary conditions.

## 3. How do you find the general solution to a PDE problem?

To find the general solution to a PDE problem, one must first identify the type of PDE (e.g. linear or nonlinear) and its order (e.g. first order or second order). Then, techniques such as separation of variables, method of characteristics, or using Green's functions can be used to solve the PDE and obtain the general solution.

## 4. Can the general solution to a PDE problem be expressed in a closed form?

In some cases, the general solution to a PDE problem can be expressed in a closed form, meaning it can be written as a finite combination of known functions. However, there are many PDE problems that do not have closed form solutions and require numerical or approximate methods to solve.

## 5. Are there applications of PDEs in real-world problems?

Yes, PDEs have numerous applications in real-world problems. For example, they are used to model heat transfer, fluid dynamics, and electromagnetic fields. They are also used in economic models and in image and signal processing. PDEs are an important tool for understanding and predicting physical phenomena in various fields.

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