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Chapter 10: Problem 20

Find the general solutions of the following differential equations: (a) \(2 x t \frac{\mathrm{d} x}{\mathrm{~d} t}=-x^{2}-t^{2}\) (b) \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=x+t \sin ^{2}\left(\frac{x}{t}\right)\) (c) \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{3 t^{2}-x^{2}}{t-2 x}\) (d) \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=x+t \tan \left(\frac{x}{t}\right)\) (e) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x+t}{x-t}\) (f) \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=x+t \mathrm{e}^{x / t}\)

Short Answer

Expert verified
General solutions are specific to each equation, typically involving constants or functions depending on the initial conditions and integration methods used.

Step by step solution

01

Step 1a: Separate Variables for Equation (a)

The given equation (a) is \(2xt \frac{\mathrm{d} x}{\mathrm{~d} t}=-x^2-t^2\). Rearrange to separate variables: \(\frac{2x}{x^2 + t^2} \frac{\mathrm{d} x}{\mathrm{~d} t} = -\frac{1}{t}\). This implies \((x + \frac{t^2}{x})\frac{\mathrm{d} x}{\mathrm{~d} t} = -t\).
02

Step 2a: Integrate Both Sides for Equation (a)

Integrate both sides of the equation separately: \(\int \frac{2x}{x^2 + t^2} \mathrm{d} x = \int -\frac{1}{t} \mathrm{d} t\). Solving these integrals gives: \(\ln|x^2 + t^2| = -\ln|t| + C\).
03

Step 3a: Solve for the General Solution of Equation (a)

Simplify the expression from the integration to find the general solution: \(x^2 + t^2 = \frac{C}{t}\).
04

Step 1b: Identify Type of Differential Equation (b)

Given \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=x+t \sin^{2}\left(\frac{x}{t}\right)\) is a first-order differential equation. Isolate \(\frac{\mathrm{d} x}{\mathrm{~d} t}\): \(\frac{\mathrm{d} x}{\mathrm{~d} t} = \frac{x}{t} + \sin^2\left(\frac{x}{t}\right)\).
05

Step 2b: Implicit Solution for Equation (b)

Notice that \(\sin^2\left(\frac{x}{t}\right)\) is periodic with respect to \(\frac{x}{t}\), suggesting a more complex solution involving \(x = A(t) + B(t) \sin^2\left(\frac{x}{t}\right)\), where \(A(t)\) and \(B(t)\) are unknown functions dependent on initial conditions.
06

Step 1c: Linearize Equation (c) Numerically

For \(t \frac{\mathrm{d} x}{\mathrm{~d} t} = \frac{3t^2-x^2}{t-2x}\), a numerical method could linearize the system into \(\frac{\mathrm{d} x}{\mathrm{~d} t} = f(t,x) = \frac{3t^2 - x^2}{t^2 - 2tx}\) for close approximations.
07

Step 1d: Simplify Trigonometric Terms in Equation (d)

For \(t \frac{\mathrm{d} x}{\mathrm{~d} t}=x+t \tan\left(\frac{x}{t}\right)\), isolate \(t \frac{\mathrm{d} x}{\mathrm{~d} t} = f(x,t) = x \sec^2\left(\frac{x}{t}\right)\), suggestive of integrating factor or approximation techniques.
08

Step 1e: Transform Variables for Equation (e)

Given \(\frac{\mathrm{d} x}{\mathrm{~d} t} = \frac{x + t}{x - t}\), this suggests using transformation \(u = x/t + 1\), \(\frac{\mathrm{d} u}{\mathrm{d} t} = \frac{2t}{x - t}\).
09

Step 1f: Expand and Simplify Equation (f)

The differential equation \(t \frac{\mathrm{d} x}{\mathrm{~d} t} = x + t \mathrm{e}^{x / t}\) can be rearranged to \(\frac{\mathrm{d} x}{\mathrm{~d} t} = \frac{x}{t} + \mathrm{e}^{x / t}\). Solving this would often involve series or expansion methods.

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Key Concepts

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

General Solution
When dealing with differential equations, finding the general solution is a primary objective. A general solution includes all possible solutions to a differential equation, typically represented with one or more arbitrary constants. This allows it to encompass various particular solutions that can satisfy initial or boundary conditions.
  • The general solution provides a robust foundation to understand the behavior of the solutions over varying conditions.
  • In the exercises given, after separating variables and integrating, we often arrive at a solution involving constants. For example, in part (a), the manipulation of the equation leads us to the general solution: \[ x^2 + t^2 = \frac{C}{t} \]
  • Such a solution highlights the integral nature and relationship between the variables involved. It also showcases the power of integrating factors or transformations when dealing with more complex forms.
Understanding how to derive the general solution is paramount as it allows the identification of specific solutions when additional conditions are applied.
First-order Differential Equations
First-order differential equations involve functions and their first derivatives. They are often expressed in the form \( \frac{dy}{dx} = f(x, y) \). These equations typically describe various real-world phenomena, from population growth to electric circuits.
  • In exercises like part (b) of the problem, identifying a differential equation as first-order suggests methods to tackle it, such as separation of variables or integration techniques.
  • Key insight for first-order equations is recognizing patterns or simplifying trigonometric or exponential terms to make them more manageable.
  • For instance, understanding periodic or trigonometric functions, as in step 1b, provides a clue on potential solution forms and complexity.
By mastering first-order differential equations, students can develop techniques to predict and model dynamic systems efficiently.
Variable Separation
Variable separation is a foundational technique for solving first-order differential equations. The idea is to rearrange an equation so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side.
  • In part (a), this method is used effectively by rearranging terms to allow for integration: \( \frac{2x}{x^2 + t^2} \frac{\mathrm{d} x}{\mathrm{~d} t} = -\frac{1}{t} \).
  • This allows both sides of the equation to be integrated separately, each with respect to its variable, leading to a solvable integral form.
  • Variable separation is particularly beneficial in equations where the relationship between the variables is not straightforward. It simplifies the problem significantly and aids in finding an integrable form.
Learning to recognize when and how to apply variable separation prepares students to handle a broad range of differential equations more confidently.
Numerical Methods
Numerical methods are vital for solving differential equations that are particularly complex or non-linear. Instead of finding an exact solution, these methods offer an approximation that is often sufficient for practical purposes.
  • For equations that cannot be solved analytically, like part (c), methods such as the Euler's method, Runge-Kutta, or iterative approaches step in to approximate solutions.
  • Numerical solutions can address real-world problems where conditions are too complex for traditional integration or algebraic methods.
  • Linearizing complex equations, as mentioned in step 1c, is a common strategy in numerical methods. It simplifies the problem into a form that can be iteratively solved, providing insights into the system's behavior.
Understanding numerical methods expands a student's toolkit for tackling differential equations, providing strategies when conventional techniques fall short.

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Most popular questions from this chapter

Denote Euler's method solution of the initial-value problem $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{x t}{t^{2}+2}, \quad x(1)=2 $$ using step size \(h=0.1\) by \(X_{\mathrm{a}}(t)\), and that using \(h=0.05\) by \(X_{\mathrm{b}}(t) .\) Find the values of \(X_{\mathrm{a}}(2)\) and \(X_{\mathrm{b}}(2) .\) Estimate the error in the value of \(X_{\mathrm{b}}(2)\), and suggest a value of step size that would provide a value of \(X(2)\) accurate to \(0.1 \%\). Find the value of \(X(2)\) using this step size. Find the exact solution of the initial-value problem, and determine the actual magnitude of the errors in \(X_{\mathrm{a}}(2), X_{\mathrm{b}}(2)\) and your final value of \(X(2)\)

What conditions on the constants \(a, b, e\) and \(f\) must be satisfied for the differential equation $$ (a x+b t) \frac{\mathrm{d} x}{\mathrm{~d} t}+e x+f t=0 $$ to be exact, and what is the solution of the equation when they are satisfied?

Find the general solutions of the following differential equations: (a) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-3 \frac{\mathrm{d} x}{\mathrm{~d} t}+4 x=\cos 4 t-2 \sin 4 t\) (b) \(9 \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-12 \frac{\mathrm{d} x}{\mathrm{~d} t}+4 x=\mathrm{e}^{-3 t}\) (c) \(2 \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+4 \frac{\mathrm{d} x}{\mathrm{~d} t}-7 x=7 \cos 2 t\) (d) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+\frac{\mathrm{d} x}{\mathrm{~d} t}+4 x=5 t-7\) (e) \(16 \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+8 \frac{\mathrm{d} x}{\mathrm{~d} t}+x=t+6\) (f) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-8 \frac{\mathrm{d} x}{\mathrm{~d} t}+16 x=-3 \sin 3 t\) (g) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-4 \frac{\mathrm{d} x}{\mathrm{~d} t}+7 x=\mathrm{e}^{-5 t}\) (h) \(3 \frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+3 \frac{\mathrm{d} x}{\mathrm{~d} t}-x=t^{2}+\mathrm{e}^{-2 t}\) (i) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 \frac{\mathrm{d} x}{\mathrm{~d} t}-3 x=5 \mathrm{e}^{-3 t}+\sin 2 t\) (j) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+16 x=1+2 \sin 4 t\) (k) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-4 \frac{\mathrm{d} x}{\mathrm{~d} t}=7-3 \mathrm{e}^{4 t}\)

Find the solutions of the following initial-value problems: (a) \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{\sin t}{x^{2}}, \quad x(0)=4\) (b) \(t^{2} \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{1}{x}, \quad x(4)=9\)

For each of the following differential equations determine whether they are exact equations and, if so, find the general solutions: (a) \(x \frac{\mathrm{d} x}{\mathrm{~d} t}+t=0\) (b) \(x \frac{\mathrm{d} x}{\mathrm{~d} t}-t=0\) (c) \((x+t) \frac{\mathrm{d} x}{\mathrm{~d} t}+x-t=0\) (d) \(\left(x-t^{2}\right) \frac{\mathrm{d} x}{\mathrm{~d} t}-2 x t=0\) (e) \((x-t) \frac{\mathrm{d} x}{\mathrm{~d} t}-x+t-1=0\) (f) \((2 x+t) \frac{\mathrm{d} x}{\mathrm{~d} t}+x+2 t=0\)
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