Question

In each of Problems 1 through 8 solve the given differential equation. $$ y^{\prime}=x^{2} / y $$

Short answer

Answer: The general solution to the differential equation \(y' = \frac{x^2}{y}\) is given by \(y(x) = \pm\sqrt{\frac{2}{3}x^3 + 2C}\), where \(C\) is an arbitrary constant.

Step-by-step solution

Rewrite the equation so that it is separable

To do this, we want to isolate \(x\) and \(y\) terms on opposite sides of the equation. In this case, we have: $$ y dy = x^2 dx $$

Integrate both sides of the equation

Now, we'll integrate both sides of the equation. For the left side, we have: $$ \int y dy $$ And for the right side, it is: $$ \int x^2 dx $$

Calculate the integrals

After calculating both integrals, we get: $$ \frac{1}{2}y^2 = \frac{1}{3}x^3 + C $$

Solving for y

To express the solution in terms of \(y\), we can multiply both sides by 2, then take the square root of both sides to isolate \(y\): $$ y^2 = \frac{2}{3}x^3 + 2C $$ $$ y = \pm\sqrt{\frac{2}{3}x^3 + 2C} $$ Thus, the general solution to the differential equation \(y' = \frac{x^2}{y}\) is given by: $$ y(x) = \pm\sqrt{\frac{2}{3}x^3 + 2C} $$

Key concepts

Understanding Integration Techniques

Separable differential equations often require integration techniques to solve them effectively. In our exercise, we began by isolating the terms involving \(y\) on one side and those involving \(x\) on the other. This is crucial for setting up the equation for integration. Here’s why:

• Left Side Integration: The expression \(\int y\, dy\) helps us find the antiderivative of \(y\), leading to \(\frac{1}{2}y^2\). This antiderivative represents the area under the curve for the function \(y\).
• Right Side Integration: For \(\int x^2\, dx\), we apply the power rule to achieve \(\frac{1}{3}x^3\). The power rule states that \(\int x^n\, dx = \frac{x^{n+1}}{n+1} + C\), which is essential for polynomial expressions.

These techniques are fundamental, allowing us to integrate and transform algebraic expressions back to their function forms, enabling the solution of differential equations.

Exploring Differential Equations

Differential equations represent relationships between functions and their derivatives. In this context, the given equation \(y' = \frac{x^2}{y}\) showcases a relationship involving the rate of change of \(y\) with respect to \(x\). Here’s how we approached solving it:

• Formulation: By translating the problem into the expression \(y\, dy = x^2\, dx\), we create a separable differential equation. This form is key because it allows direct integration.
• Solution Process: Solving differential equations typically involves finding an unknown function that satisfies the equation. Here, integration helps us trace back the derivative to find \(y(x)\).

Differential equations are a cornerstone of describing various natural phenomena, modeling everything from population growth to physical systems' behaviors in engineering.

Finding General Solutions

The general solution of a differential equation provides a family of functions containing arbitrary constants that satisfy the equation. In our scenario, we arrived at \(y(x) = \pm\sqrt{\frac{2}{3}x^3 + 2C}\).

• Role of the Constant \(C\): The integration gives rise to an arbitrary constant \(C\), signifying a family of solutions. Different values of \(C\) represent different particular solutions.
• Understanding \(y(x)\): The \(\pm\) sign indicates that both positive and negative square roots are possible solutions, meaning that for the same \(x\), \(y\) could be positive or negative.

General solutions are comprehensive and can be adapted to specific initial conditions for particular solutions, forming the basis for solving real-world problems where exact values are needed.