Question

Find the general solution of each differential equation. Try some by calculator. $$x d y=\left(3 x^{2}+y\right) d x$$

Short answer

Solving this differential equation requires further analysis and may not be directly integrable using simple methods. It requires identification of the correct method, potentially involving an integrating factor, substitution, or recognizing it as an exact differential equation.

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is in non-standard form. Rewrite it to identify its type. In this case, it can be identified as a first-order non-linear differential equation.

Rewrite the Differential Equation in the Standard Form

The standard form for a first-order Ordinary Differential Equation (ODE) is \(\frac{dy}{dx} = f(x, y)\). Rewrite the equation in this form by dividing both sides by \(x\) to get \(\frac{dy}{dx} = 3x + \frac{y}{x}\).

Separate the Variables if Possible

To separate variables, manipulate the equation so that all terms with \(y\) are on one side, and all terms with \(x\) are on the other. In this equation, the term \(\frac{y}{x}\) complicates separation, indicating this might not be a separable equation.

Check for an Exact or Homogeneous Differential Equation

For an equation of the form \(Mdx + Ndy = 0\), check if \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\) and if the equation is homogeneous. In our case, let's set \(M = y\) and \(N = 3x^2 + y\) and check for exactness or homogeneity.

Solve if Homogeneous

If the equation is homogeneous, use the substitution \(v = \frac{y}{x}\) and solve the resulting differential equation. After solving for \(v\), integrate to find \(y\) as a function of \(x\).

Check for Other Methods or Integrate Directly

If the equation is not separable, exact, or homogeneous, check if it can be integrated directly or if another method, like integrating factors or substitution, is required. Since the equation is not easily integrated directly, further analysis to determine the correct method would be needed for a complete solution.

Key concepts

Ordinary Differential Equation

An ordinary differential equation (ODE) is a relation involving functions of only one independent variable and its derivatives. The term 'ordinary' is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. ODEs play a crucial role in various scientific and engineering disciplines as they describe the dynamics of systems ranging from simple mechanical systems to complex biological networks.

ODEs are classified by order, determined by the highest derivative present. First-order ODEs, like the one presented in our exercise \(x dy = (3x^2 + y) dx\), involve only the first derivative of the function. This exercise asks for finding the general solution, which means determining the function \(y(x)\) that satisfies the equation for all \(x\).

Part of solving an ODE like ours involves rewriting it into a standard form to identify the type and applying suitable solving methods, which could range from separation of variables to more complex techniques, depending on the nature of the equation.

Separable Variables

The method of separable variables is a simple, yet powerful technique for solving certain types of first-order ODEs. It’s based on the idea of separating the variables so that each side of the equation contains only one variable—either \(x\) or \(y\).

To apply this method, manipulate the equation to express it in the form \(g(y)dy = f(x)dx\), where \(g(y)\) and \(f(x)\) are functions of \(y\) and \(x\) respectively. After separating variables, integrate both sides to obtain the solution. If the equation cannot be easily rewritten to this form, as in \(\frac{dy}{dx} = 3x + \frac{y}{x}\) from our original problem, the equation is not separable, and other methods must be considered.

Homogeneous Differential Equation

Homogeneous differential equations have a very specific definition. A first-order ODE in the form \(dy/dx = f(y/x)\) or adaptable into this form, where the function \(f\) is homogeneous of degree 0, is considered to be a homogeneous differential equation. Homogeneous functions satisfy the property \(f(ty, tx) = f(y, x)\) for all \(t\).

For the ODE at hand from the exercise \(x dy = (3x^2 + y) dx\), after setting \(M = y\) and \(N = 3x^2 + y\) the check for homogeneity involves examining if the terms can be expressed as functions of \(v = y/x\). This allows for substitution, which then leads to an equation with separable variables. Following this, solving and then reverting the substitution should yield the relation between \(y\) and \(x\) that solves the original ODE.