Question

$$ \left(x \tan \frac{y}{x}+y\right) d x-x d y=0 $$

Short answer

The general solution for the given differential equation is: \[y = x(1 - e^{C_1x + \ln{x}})\] where \(C_1\) is an integration constant.

Step-by-step solution

Rewrite the differential equation

Let's write the equation in the form \(M(x, y) dx + N(x, y) dy = 0\): \[(x\tan{(\frac{y}{x})} + y)dx - xdy = 0\]

Define M(x, y) and N(x, y)

From the differential equation, we can define: M(x, y) = \(x\tan{(\frac{y}{x})} + y\) N(x, y) = -x

Find substitution

Notice that x and y always appear in the form \(\frac{y}{x}\). Let's make a substitution: \(v = \frac{y}{x}\) \(y = vx\) Now, we differentiate both sides of \(y=vx\) with respect to x to find the relationship between dy and dv: \[\frac{dy}{dx} = v + x\frac{dv}{dx}\]

Replace y and dy in the differential equation

Now, we'll substitute y and dy with the new expressions in terms of x and v: \[(x\tan{(v)} + vx)dx - x(v + x\frac{dv}{dx})dx = 0\]

Solve for v

Now, we solve the simplified equation: \[(x\tan{(v)} + vx) - x(v + x\frac{dv}{dx}) = 0\] Let's rearrange the terms: \(x\tan{(v)}(1-v) = xv(1+\frac{dv}{dx})\) Now, we can cancel the x factor and simplify further: \[\tan(v)(1-v) = v(1+\frac{dv}{dx})\]

Separate the variables

We will now separate the variables: \[\frac{1-v}{v} dv = \frac{dx}{x}\]

Integrate both sides

Now integrate both sides to get: \[\int \frac{1-v}{v} dv = \int \frac{1}{x} dx\] The next step is to perform the integration: \[-\ln{|v|} + \ln{|1-v|} = \ln{|x|} + C_1\]

Solve for y

Now substitute back the original variables to get the general solution: \[-\ln{|\frac{y}{x}|} + \ln{|1-\frac{y}{x}|} = \ln{|x|} + C_1\] Finally, solve for y to get the general solution for the given differential equation: \[y = x(1 - e^{C_1x + \ln{x}})\] Here, the value of \(C_1\) can be found by using the initial conditions (if given), and otherwise, the solution will remain in terms of the constant \(C_1\).

Key concepts

Variable Separable Method

The Variable Separable Method is a simple yet powerful technique used to solve certain types of ordinary differential equations. This method relies on the idea that differentials can be separated when the equation can be rearranged in a form where all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side. This format allows each side to be integrated separately.

When applying this technique, we're essentially dividing the problem into two parts:

• The first part involves rearranging the equation such that all 'dy' terms are on one side and all 'dx' terms are on the other. This step is crucial as it sets the stage for the integration process that follows.
• The second part involves integrating both sides. The integration leads us to a solution that might involve an integration constant, often denoted as 'C'. This constant captures the infinite possibilities of situations meeting the conditions of the differential equation.

The variable separable method is especially handy when the equation exhibits a structure that naturally lends itself to separation.

Substitution Method

The Substitution Method in solving differential equations involves choosing a substitution that simplifies the original equation into a more manageable form. This method is particularly useful in tackling equations that are not amenable to the variable separable method.

In the given solution, we used the substitution \( v = \frac{y}{x} \). This substitution effectively transforms the given differential equation, where the variables x and y appear in the form of the ratio \( \frac{y}{x} \). This presents two advantages:

• It reduces the equation's complexity, essentially lowering a higher-order problem into a simpler one to deal with.
• It turns a difficult differential equation into another form that is more conducive to integration or further simplification.

Once the substitution is made, the new variable brings the equation to more familiar grounds, allowing us to use standard techniques to find solutions.

Integration Techniques

In solving differential equations using both the variable separable and substitution methods, integration plays a crucial role. Once we have successfully separated variables or made an effective substitution, we often arrive at an equation that needs to be integrated.

Some integration techniques commonly used include:

• Simplification: This involves rewriting the integrand in a form that is easier to integrate, often by factoring or canceling terms when possible.
• Using Known Integrals: Relying on fundamental integral formulas for known functions, such as \( \int \frac{1}{x}dx = \ln{|x|} \) or \( \int e^x dx = e^x + C \).
• Integration by Parts or Substitution: These are additional techniques applied for more complicated functions, where simple integration isn't sufficient.

In our original exercise, the final integration resulted in leveraging natural logarithms and arithmetic manipulations to simplify the transformed equation to achieve the general solution. Proper understanding of integration laws and methods is crucial for mastering ordinary differential equations.