Question

The solution of the differential equation \(\frac{d y}{d x}=\frac{x^{2}+x y+y^{2}}{x^{2}}\) (1) \(\tan ^{-1}\left(\frac{x}{y}\right)=\ell n y+c\) (2) \(\tan ^{-1}\left(\frac{y}{x}\right)=\ell n y+c\) (3) \(\tan ^{-1}\left(\frac{y}{x}\right)=\ell n x+c\) (4) \(\sin ^{-1}\left(\frac{y}{x}\right)=\ln x+c\)

Short answer

The correct solution is (3) arctan(y/x) = ln x + C.

Step-by-step solution

Separate the variables

Rewrite the given differential equation: dy/dx = (x^2 + xy + y^2) / x^2 Separate the variables to make it easier to integrate: (dy/dx) = (1 + y/x + (y/x)^2)

Use substitution

Let v = y/x, then y = vx and dy/dx = v + x dv/dx.Rewrite the equation in terms of v and x: v + x dv/dx = 1 + v + v^2.

Simplify the equation

Simplify the equation:x dv/dx = v^2 + 1.Rearrange to separate variables: dv / (v^2 + 1) = dx / x.

Integrate both sides

Integrate both sides of the equation:∫dv / (v^2 + 1) = ∫dx / x.This gives:arctan(v) = ln|x| + C.

Substitute v back in terms of y and x

Substitute v = y/x back into the equation:arctan(y/x) = ln|x| + C.

Key concepts

Separation of Variables

Separation of variables is a powerful technique used to solve differential equations. It involves rearranging the equation to isolate the dependent and independent variables on opposite sides. This method simplifies integration, making the solution process manageable.
To separate variables in the given differential equation \(\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\), we rewrite it as \(\frac{dy}{dx} = 1 + \frac{y}{x} + \frac{(y/x)^2}\). This new form allows us to integrate both sides separately after substitution.

Substitution Method

The substitution method is used to simplify differential equations by introducing a new variable. This often makes the equation easier to solve. In this example, we use the substitution \(v = \frac{y}{x}\). This transforms the variables, resulting in \(y = vx\) and \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).
By rewriting the original equation using \(v\), we get \(v + x\frac{dv}{dx} = 1 + v + v^2\). This step highlights the convenience of the substitution method in simplifying complex relationships.

Integration Techniques

Integration techniques are essential for solving separated probability problems after employing substitution. Once we rearrange the equation \(x \frac{dv}{dx} = v^2 + 1\) to \(\frac{dv}{v^2 + 1} = \frac{dx}{x}\), we are ready to integrate both sides.
Integrating the left side, \(\frac{dv}{v^2 + 1}\), we recognize it as a standard integral, resulting in the arctangent function \(\text{arctan}(v)\). On the right side, integration of \( \frac{dx}{x}\) yields the natural logarithm function, \( \text{ln}|x| + C\).

Arctangent Function

The arctangent function, \(\text{arctan}(v)\), emerges frequently in integrals involving \(\frac{1}{1+v^2}\). During our integration step, we find \( \text{arctan}(v) = \text{ln}|x| + C\).
This function is particularly useful in trigonometric integrals and helps to connect the solution back to the initial variables. By substituting \(\frac{y}{x}\) for \(v\), we get the final solution equation for our differential equation: \(\text{arctan}\frac{y}{x} = \text{ln}|x| + C\).

Natural Logarithm

The natural logarithm, denoted \(\text{ln}(x)\), is crucial in calculus and differential equations. It arises naturally when integrating \( \frac{dx}{x}\), providing solutions that often involve exponentiation. In our example, the integration results in \(\text{ln}|x| + C\).
The use of \(\text{ln}(x)\) ensures that the solution adequately represents the relationship between variables after integration. It captures the exponential decay or growth behavior inherent in many differential problems.