Question

For the following problems, find the general solution to the differential equation.\(x^{\prime}=t \sqrt{4+t}\)

Short answer

The general solution is \( x(t) = \frac{2}{5}(4+t)^{5/2} - \frac{4}{3}(4+t)^{3/2} + C \).

Step-by-step solution

Understand the Problem

The differential equation given is \( x' = t \sqrt{4+t} \). The goal is to find the general solution for the variable \( x \). To do this, we need to integrate the right-hand side with respect to \( t \).

Set up the Integration

The equation \( x' = t \sqrt{4+t} \) implies \( \frac{dx}{dt} = t \sqrt{4+t} \). To solve for \( x(t) \), we rearrange the terms to write the integral: \( x(t) = \int t \sqrt{4+t} \, dt \).

Simplify the Integrand

The integrand is \( t \sqrt{4+t} \). To simplify, use a substitution method. Let \( u = 4+t \), which means \( du = dt \) and \( t = u - 4 \). Substitute these into the integral: \( \int (u-4) \sqrt{u} \, du \).

Integrate Using Substitution

Now integrate the function \((u-4)\sqrt{u}\) by evaluating two separate integrals: \( \int u^{3/2} du - 4 \int u^{1/2} du \). The antiderivatives are \( \frac{2}{5}u^{5/2} \) and \( \frac{4}{3}u^{3/2} \), respectively.

Solve the Integral

The integration yields: \( \frac{2}{5}u^{5/2} - \frac{4}{3}u^{3/2} + C \). Substitute back \( u = 4 + t \) to return to the original variable: \( \frac{2}{5}(4+t)^{5/2} - \frac{4}{3}(4+t)^{3/2} + C \).

Write the General Solution

The general solution to the differential equation is thus \( x(t) = \frac{2}{5}(4+t)^{5/2} - \frac{4}{3}(4+t)^{3/2} + C \), where \( C \) is the constant of integration.

Key concepts

Integration Techniques

In differential equations, integration is an essential tool used to solve for unknown functions. It involves finding the antiderivative of a given function. When faced with a differential equation like \(x' = t \sqrt{4+t}\), one must integrate its right-hand side to find the expression for \(x(t)\). One of the primary techniques used in integration is handling complex expressions through simplification or transformation, such as substitution or integration by parts.
This process might also involve splitting integrals or using standard antiderivative formulas. The objective is to manipulate the integrand to a form that allows easy integration. Mastery of these techniques increases one's ability to solve diverse and challenging mathematical problems. By practicing various integration methods, you open yourself to more strategic problem-solving approaches in calculus, ensuring effective solutions.

Substitution Method

The substitution method is a vital technique in integration for simplifying complex functions. In this process, you replace a part of the integrand with a single variable, making the integration more straightforward. For our problem, \( u = 4 + t \) was chosen as the substitution. This transforms \( t \sqrt{4+t} \) into a simpler expression, unlocking easier integration.

This technique is useful when:
• The integrand embodies composition (a function within a function).
• The derivative of the substitution variable can simplify the integrand.
• Making the substitution results in a standard formula for integration.

In our example, the transformation allows us to handle the expression \((u-4)\sqrt{u}\), which is much more manageable. Substitution not only streamlines the process but can also reveal insights into the structure of the integrand benefiting further integration.

General Solution

Finding the general solution of a differential equation means determining a function that satisfies the equation over a set of conditions. It represents a family of functions rather than a specific one, showcasing the range of possibilities based on varying constants.
For the equation \(x' = t \sqrt{4+t}\), through integration and back substitution, we obtained:\[ x(t) = \frac{2}{5}(4+t)^{5/2} - \frac{4}{3}(4+t)^{3/2} + C \]

In this context, the general solution:
• Accounts for all possible initial conditions.
• Includes an arbitrary constant \( C \), which will adjust to fit particular cases.
• Is essential in modeling real-world phenomena when basic conditions are not predefined.

Understanding the general solution is critical because it provides not only a basic framework but also reveals how specific solutions can be sculpted from a broader structure by modifying constants.

Constant of Integration

The constant of integration, denoted as \(C\), emerges when calculating indefinite integrals. It reflects the infinite spectrum of antiderivatives differing by a constant. Why do we need it? Because when differentiating, constants disappear, making it necessary to reintroduce them during integration.

Purpose and significance of \(C\):
• Helps in exhibiting all potential antiderivatives for a function.
• Ensures completeness, accommodating initial conditions or boundaries.
• Allows personalized solutions based on additional given data or constraints in applied problems.

In our solution for \(x(t) = \frac{2}{5}(4+t)^{5/2} - \frac{4}{3}(4+t)^{3/2} + C\), \( C \) embodies the flexibility needed to tailor the solution. In practical applications, \( C \) is computed by substituting specific initial values into the equation. This makes the theoretically infinite solutions tangible and applicable to real-world scenarios.