Question

Solve the given differential equations. $$ \frac{d x}{d t}+\frac{t+1}{2 t} x=\frac{t+1}{x t} $$

Short answer

The general solution for the given first-order ordinary differential equation is: \[x(t) = \frac{1}{e^{\frac{1}{2}(t + \ln|t|)}}\left(\int e^{\frac{1}{2}(t + \ln|t|)}\frac{t+1}{xt} dt + C\right)\]

Step-by-step solution

Identify the ODE and Write it in Standard Form

We are given the first-order ODE: \[ \frac{dx}{dt} + \frac{t+1}{2t}x = \frac{t+1}{xt} \]

Find the Integrating Factor

To find the integrating factor, we consider the coefficient of the x term in the ODE, which is \(\frac{t+1}{2t}\). The integrating factor, \(I(t)\), is calculated using the following formula: \[ I(t) = e^{\int \frac{t+1}{2t} dt} \] Now, let's calculate the integral: \[ \begin{aligned} \int \frac{t+1}{2t} dt &= \frac{1}{2}\int \frac{t}{t} + \frac{1}{t} dt \\ &= \frac{1}{2}\int 1 + \frac{1}{t} dt \\ &= \frac{1}{2}(t + \ln|t|) + C_1 \end{aligned} \] Since we only need I(t), we do not need to find the constant C1, so the integrating factor I(t) is: \[ I(t) = e^{\frac{1}{2}(t + \ln|t|)} \]

Multiply the ODE by the Integrating Factor

Now we multiply the entire differential equation by the integrating factor I(t): \[ e^{\frac{1}{2}(t + \ln|t|)}\left(\frac{dx}{dt} + \frac{t+1}{2t}x\right) = e^{\frac{1}{2}(t + \ln|t|)}\frac{t+1}{xt} \] This will make the left side of the equation an exact differential.

Integrate both Sides of the Equation

Now we integrate both sides of the equation with respect to t: \[ \begin{aligned} \int \frac{d(xI(t))}{dt} dt &= \int e^{\frac{1}{2}(t + \ln|t|)}\frac{t+1}{xt} dt \\ xI(t) &= \int e^{\frac{1}{2}(t + \ln|t|)}\frac{t+1}{xt} dt + C \end{aligned} \]

Solve for x(t)

To find the solution x(t), we need to isolate x on one side of the equation. We do this by dividing both sides by the integrating factor I(t): \[ x(t) = \frac{1}{e^{\frac{1}{2}(t + \ln|t|)}}\left(\int e^{\frac{1}{2}(t + \ln|t|)}\frac{t+1}{xt} dt + C\right) \] This is the general solution for the given first-order ordinary differential equation.

Key concepts

Integrating Factor Method

The integrating factor method is a powerful tool for solving first-order ordinary differential equations (ODEs), especially when they are not already exact. This method involves multiplying the entire ODE by an 'integrating factor' that is chosen specifically to make the left side of the equation become the derivative of some function.

For the given problem, we identify an integrating factor by looking at the coefficient of the term which is a function of the independent variable, t. Here, the integrating factor, denoted as I(t), is found through the integral \[I(t) = e^{\int \frac{t+1}{2t} dt}\]. Once the integrating factor is determined, we multiply it across the ODE to transform it into an easily integrable form. The exact differential equation that results allows us to integrate directly and find a solution for the dependent variable, in this case, x(t).

The beauty of this method lies in its ability to tackle equations that do not initially seem solvable by straightforward integration. Through the systematic application of the integrating factor, the equation's structure is manipulated into a more friendly form, leading to a solution.

First-Order ODE

First-order ordinary differential equations, or first-order ODEs, are equations that involve a function and its first derivative. They are fundamental in the mathematical modeling of dynamic systems, capturing how a system evolves with one independent variable, which is often time. The general form of a first-order ODE can be expressed as \[ \frac{dx}{dt} = f(t, x) \], where f(t, x) is some given function of t and x.

In the exercise provided, we have a slightly more complex version where the ODE is nonlinear and involves the function and its derivative, in addition to the product of the function. To solve such equations, methods like separation of variables, integrating factor, or substitution might be used, depending on the form of the equation. The integrating factor method, applied in this exercise, transforms a non-exact ODE into an exact differential equation by multiplying it by an appropriately chosen function of t.

Exact Differential Equation

An exact differential equation is one where the left-hand side can be expressed as the derivative of a product of two functions. Specifically, when we say that a first-order ODE is exact, it means that the equation can be written in the form \[ \frac{d(xI(t))}{dt} = g(t, x) \], such that \[xI(t)\] can be obtained by direct integration.

In the context of our exercise, after applying the integrating factor, we rearrange the equation so that one side becomes the differential of a product. This is a crucial step because it signals that we can integrate both sides with respect to t to find a more direct path to the solution. The exactness condition essentially guarantees that the procedure of multiplication by the integrating factor will result in an expression where the integration process is straightforward. This approach simplifies the solving of many first-order ODEs, which would otherwise require much more complex methods to solve.