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Chapter 7: Problem 11

\(9-14=\) Find the solution of the differential equation that satisfies the given initial condition. $$\frac{d u}{d t}=\frac{2 t+\sec ^{2} t}{2 u}, \quad u(0)=-5$$

Short Answer

Expert verified
The solution is \(u = -\sqrt{t^2 + \tan t + 25}\).

Step by step solution

01

Separate the variables

Start by separating the variables in the differential equation: \( \frac{d u}{d t} = \frac{2 t + \sec^2 t}{2 u} \). Multiply both sides by \(2u\) to get \(2u \frac{d u}{d t} = 2t + \sec^2 t\). Then, divide through by \(2t + \sec^2 t\) and multiply through by \(d t\) to get \(2u \, d u = (2t + \sec^2 t) \, d t\).
02

Integrate both sides

Integrate both sides of the equation. The left side becomes \(\int 2u \, d u = u^2 + C_1\), and the right side becomes \(\int (2t + \sec^2 t) \, d t = t^2 + \tan t + C_2\). Thus, the integral gives \(u^2 + C_1 = t^2 + \tan t + C_2\).
03

Consolidate the constants

Combine the constants to simplify the expression. Let \(C = C_2 - C_1\), so the equation becomes \(u^2 = t^2 + \tan t + C\).
04

Apply the initial condition

Use the initial condition \(u(0) = -5\) to find \(C\). Substituting \(t = 0\) and \(u = -5\) gives \((-5)^2 = 0^2 + \tan(0) + C\). This simplifies to \(25 = 0 + 0 + C\), so \(C = 25\).
05

Rewrite the particular solution

Substitute \(C = 25\) back into the equation to get the particular solution: \(u^2 = t^2 + \tan t + 25\). Solve for \(u\) to get \(u = \pm \sqrt{t^2 + \tan t + 25}\). Since the initial condition specifies \(u(0) = -5\), choose the negative root: \(u = -\sqrt{t^2 + \tan t + 25}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Conditions
When solving a differential equation, initial conditions are crucial. They give us specific values for the variables at a particular point, which helps pinpoint a unique solution among countless possibilities. In our problem, the initial condition provides the value of the unknown function when the independent variable is zero:
  • The function value is given as \( u(0) = -5 \).
  • This means at time \( t = 0 \), the dependent variable \( u \) equals \( -5 \).
This specific point becomes our guiding start. It ensures the solution we find fits perfectly with the scenario described. By applying this fixed point later in the solving process, we determine the constant of integration that makes the differential equation solution unique.
Separation of Variables
Separation of variables is a method used for solving differential equations, where the idea is to rearrange the equation such that each variable and its respective derivative are isolated on different sides of the equation.
  • In our problem, the differential equation \( \frac{d u}{d t} = \frac{2 t + \sec^2 t}{2 u} \) needs to be rearranged.
  • We multiply both sides by \( 2u \), resulting in \( 2u \frac{d u}{d t} = 2t + \sec^2 t \).
  • Divide through by \( 2t + \sec^2 t \) and multiply through by \( dt \).
This yields \( 2u \, du = (2t + \sec^2 t) \, dt \). Now, each side depends solely on either \( u \) or \( t \), allowing for straightforward integration.
Integration
Integration is a fundamental calculus operation used to find the integral or anti-derivative, which is essentially the inverse of differentiation. After separating the variables, the equation is ready for integration:
  • The left side becomes \( \int 2u \, d u \), which upon integration gives \( u^2 + C_1 \).
  • The right side \( \int (2t + \sec^2 t) \, d t \) results in \( t^2 + \tan t + C_2 \) upon integration.
Combining these integrals, we have: \( u^2 + C_1 = t^2 + \tan t + C_2 \). The constants \( C_1 \) and \( C_2 \) arise due to the indefinite integration, but they can be combined into a single constant \( C = C_2 - C_1 \). The next step is to simplify this equation further by applying the initial conditions.
Definite Integration
In this context, definite integration refers to finding the specific solution to a differential equation by employing the initial conditions to determine the constant of integration.
  • We have the equation \( u^2 = t^2 + \tan t + C \).
  • Applying the initial condition \( u(0) = -5 \) simplifies the expression.
By setting \( t = 0 \) and \( u = -5 \), we solve for \( C \) as follows: \[-5^2 = 0^2 + \tan 0 + C\] This results in \( 25 = C \). Thus, the revised equation is \( u^2 = t^2 + \tan t + 25 \). Finally, solve for \( u \), taking into account the initial condition, leading us to \( u = -\sqrt{t^2 + \tan t + 25} \). This specific solution honors the analysis rooted in our initial condition, sealing it with the real-world context provided.

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