Question

Solve the given differential equation. $$ \frac{d y}{d x}=\frac{x^{2}}{1+y^{2}} $$

Short answer

Question: Solve the given differential equation and find the implicit solution for the function 'y': $$\frac{dy}{dx} = \frac{x^2}{1 + y^2}$$ Solution: After solving the given differential equation using the separation of variables technique, we found the implicit solution as follows: $$y + \frac{y^3}{3} - \frac{x^3}{3} = C$$

Step-by-step solution

Separate the variables

Rewrite the given equation as follows: $$ \frac{dy}{dx} = \frac{x^2}{1 + y^2} $$ Now, let's separate the variables by moving the 'y' terms on one side and the 'x' terms on the other side. To do so, we simply multiply both sides by \(\(1 + y^2\)\) and \(\(dx\)\) to get: $$ (1 + y^2) dy = x^{2} dx $$

Integrate both sides

Integrate both sides with respect to their respective variables: $$ \int (1 + y^2) dy = \int x^{2} dx $$ The integration of the left side is quite simple, just integrate '1' and '\(\(y^2\)\)', meanwhile, integrate 'x^2' on the right side.

Evaluate the integrals

Left-hand side integral: $$ \int (1 + y^2) dy = y + \frac{y^3}{3} + C_1 $$ Right-hand side integral: $$ \int x^{2} dx = \frac{x^3}{3} + C_2 $$ Now, equate both sides: $$ y + \frac{y^3}{3} + C_1 = \frac{x^3}{3} + C_2 $$

Simplify the equation

To simplify the equation we will combine the constants and set a new constant '\(\(C\)\)' $$ y + \frac{y^3}{3} - \frac{x^3}{3} = C $$ At this point, we have obtained a solution for the given differential equation. It is an implicit solution since the function 'y' has not been explicitly solved. However, obtaining an explicit solution from this point is typically not possible, so the implicit solution is generally accepted. The final solution is: $$ y + \frac{y^3}{3} - \frac{x^3}{3} = C $$

Key concepts

Integration Techniques

Integration is a fundamental technique used to find solutions to differential equations. When solving separable differential equations, integration enables us to handle expressions involving differentials by evaluating the antiderivative of a function. In our example, after separating the variables, we obtain two integrals that need evaluation:

• \( \int (1 + y^2) \, dy \)
• \( \int x^2 \, dx \)

Each integral is approached individually. For \(\int (1 + y^2) dy\), we treat "1" and "\(y^2\)" as separate functions and integrate them independently, yielding \( y + \frac{y^3}{3} \). Likewise, for \(\int x^2 \, dx \), we find the antiderivative to be \(\frac{x^3}{3}\).
Integration requires adding a constant of integration, \(C\), to account for indefinite integrals which represent families of functions. In practice, integration techniques are crucial because they transform the problem of solving differential equations into a more accessible form by evaluating key parts of the equations.

Separation of Variables

Separation of variables is a technique used to solve differential equations by separating the variables (i.e., dependent and independent variables) onto different sides of the equation. This technique is particularly useful for separable differential equations where the expression can be manipulated into a form \(N(y)\,dy = M(x)\,dx\).
In the given exercise, we first rewrote the equation \(\frac{dy}{dx} = \frac{x^2}{1 + y^2}\) in a separated form \((1 + y^2) \, dy = x^2 \, dx\). By adjusting the equation in such a way, each side contains only one variable and its differential.
This clear separation allows us to integrate each side easily with respect to its variable, leading directly to the solution without the variables being mixed. The goal is to manipulate the equation so that it can be integrated comfortably. This method is highly valued because of its direct approach and simplicity in handling certain types of differential equations.

Implicit Solutions

An implicit solution is a form of the solution to a differential equation where the dependent variable is not isolated on one side of the equation. Instead, the solution involves both the dependent and independent variables mixed together within the equation.
In our solution, the implicit form \(y + \frac{y^3}{3} - \frac{x^3}{3} = C\) is derived. The presence of terms involving both \(y\) and \(x\) indicates this implicit nature. The \(C\) in the equation represents a constant of integration which becomes part of the implicit solution.

• Implicit solutions arise when it is not feasible to solve explicitly for the dependent variable \(y\).
• They are accepted as valid solutions, particularly for more complex differential equations.

Implicit solutions are valuable because they still describe the relationship between the variables, even if not in a straightforward, explicit form. In many cases, obtaining an explicit solution might not be possible, making implicit solutions a necessary aspect of solving differential equations.