Question

Find the solution to the initial-value problem.\(y^{\prime}=3 x^{2}\left(y^{2}+4\right), y(0)=0\)

Short answer

The solution is \(y = 2 \tan(2x^3)\).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \(y' = 3x^2(y^2 + 4)\). Notice that it is a first order differential equation since it involves \(y'\) and not higher derivatives.

Separate Variables

This differential equation is separable. Separate the variables by rewriting it as \(\frac{dy}{y^2 + 4} = 3x^2 \, dx\). This allows us to integrate both sides separately with respect to their respective variables.

Integrate Both Sides

Integrate both sides: on the left side, the integral becomes \(\int \frac{dy}{y^2 + 4}\), which is an arctan function, and on the right side \(\int 3x^2 \, dx\) gives a polynomial.\[\int \frac{dy}{y^2 + 4} = \frac{1}{2} \tan^{-1}(\frac{y}{2}) + C_1\]\[\int 3x^2 \, dx = x^3 + C_2\]

Combine and Solve for y

Combine the results from the integrations:\[ \frac{1}{2} \tan^{-1}\left(\frac{y}{2}\right) = x^3 + C \]By multiplying both sides by 2, you can further simplify and solve for \(y\):\[ \tan^{-1}\left(\frac{y}{2}\right) = 2x^3 + C \]

Apply the Initial Condition

Use the initial condition \(y(0) = 0\) to find the constant \(C\). Substitute \(x = 0\) and \(y = 0\) into the equation:\[ \tan^{-1}\left(\frac{0}{2}\right) = 2(0)^3 + C \]This simplifies to \(0 = C\). Thus, \(C = 0\).

Find the Solution

Substitute \(C = 0\) back into the equation:\[ \tan^{-1}\left(\frac{y}{2}\right) = 2x^3 \]To solve for \(y\), take the tangent of both sides:\[ \frac{y}{2} = \tan(2x^3) \]Finally, multiply by 2 to isolate \(y\):\[ y = 2 \tan(2x^3) \]

Key concepts

Separable Differential Equations

A separable differential equation is one that can be rewritten so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the other side. This characteristic makes them quite easy to solve as it allows each side to be integrated independently.

To separate a given differential equation, like the one in the exercise, we start by rewriting the equation. In this case, the equation is given as: \[ y' = 3x^2(y^2 + 4) \]
By expressing this in a way where each variable is isolated with its differential, we get:\[ \frac{dy}{y^2 + 4} = 3x^2 \, dx \]
This rearrangement means that the problem can be solved by performing definite integrals on each side independently. With the variables successfully separated, the equation can then be solved through the appropriate integration methods, simplifying the process significantly.

First Order Differential Equation

When we talk about first order differential equations, we refer to equations involving the first derivative of an unknown function, usually denoted as \(y'\), and no higher derivatives. The given equation, \(y' = 3x^2(y^2 + 4)\), demonstrates this by only involving the first derivative \(y'\).

First order differential equations can describe a wide range of phenomena. They are pivotal in modeling rates of change in many fields, like physics, biology, and economics. Solving them often involves finding a function \(y\) that adheres to the differential relationship provided. Some of these equations are linear, but many are nonlinear like the one in the example. Interestingly, the method of separation of variables, as used earlier, can be a major tool in solving such nonlinear equations when they are separable.

Solutions to first order differential equations provide us with functions that describe how a quantity affected by the factors in the equation evolves over time or another variable context.

Integration Methods

Integration is the centerpiece of solving separable differential equations. Once an equation is separated, we rely heavily on integration methods to find the solution. The goal is to take the integral of both sides of the equation after separation.

In the given exercise, the left side of the separated equation involves an arctangent integration \[ \int \frac{dy}{y^2 + 4} = \frac{1}{2} \tan^{-1}\left(\frac{y}{2}\right) + C_1 \]
This represents a typical case where recognizing standard integral forms is beneficial. For the right side:\[ \int 3x^2 \, dx = x^3 + C_2 \]
Here we use basic polynomial integration. Understanding these forms allows for straightforward integration, turning differential equations into solvable expressions. Both sides are integrated independently, yielding a result that can be combined, ensuring the complete function matches given initial conditions.

Thus, knowing how to handle various integration techniques becomes essential for tackling first order and separable differential equations effectively, leading to a specified final solution.