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Chapter 10: Problem 75

Find the value of \(X(0.3)\) for the initial-value problem $$ \begin{aligned} &\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+x^{2} \frac{\mathrm{d} x}{\mathrm{~d} t}+x=\sin t \\ &x(0)=0, \quad \frac{\mathrm{d} x}{\mathrm{~d} t}(0)=1 \end{aligned} $$ using Euler's method with step size \(h=0.1\).

Short Answer

Expert verified
The value of \( X(0.3) \) is approximately 0.300998.

Step by step solution

01

Understand the Initial Value Problem

We need to solve the second-order differential equation \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+x^{2} \frac{\mathrm{d}x}{\mathrm{~d} t}+x=\sin t\) subjected to the initial conditions \(x(0)=0\) and \(\frac{\mathrm{d} x}{\mathrm{~d} t}(0)=1\) using Euler's method. We will approximate the solution up to \( t = 0.3 \) with a step size of \( h = 0.1 \).
02

Convert to a System of First-Order Equations

Introduce the variables \( y_1 = x \) and \( y_2 = \frac{\mathrm{d} x}{\mathrm{~d} t} \). This converts the second-order equation into a system:\[\frac{\mathrm{d} y_1}{\mathrm{~d} t} = y_2\]\[\frac{\mathrm{d} y_2}{\mathrm{~d} t} = \sin t - y_1 - y_1^2 y_2 \]
03

Set Initial Conditions

At \( t = 0 \), the initial conditions are \( y_1(0) = 0 \) and \( y_2(0) = 1 \).
04

Apply Euler's Method for \( t = 0.1 \)

Using Euler's method, approximate the values of \( y_1 \) and \( y_2 \) at \( t = 0.1 \):\[ y_1(0.1) = y_1(0) + h \times y_2(0) = 0 + 0.1 \times 1 = 0.1 \]\[ y_2(0.1) = y_2(0) + h \times (\sin(0) - y_1(0) - y_1(0)^2 \times y_2(0)) = 1 + 0.1 \times (0 - 0) = 1 \]
05

Apply Euler's Method for \( t = 0.2 \)

Calculate \( y_1 \) and \( y_2 \) at \( t = 0.2 \):\[ y_1(0.2) = y_1(0.1) + h \times y_2(0.1) = 0.1 + 0.1 \times 1 = 0.2 \]\[ y_2(0.2) = y_2(0.1) + h \times (\sin(0.1) - y_1(0.1) - y_1(0.1)^2 \times y_2(0.1)) = 1 + 0.1 \times (0.0998 - 0.1 - 0.01) = 1.00998 \]
06

Apply Euler's Method for \( t = 0.3 \)

Calculate \( y_1 \) and \( y_2 \) at \( t = 0.3 \):\[ y_1(0.3) = y_1(0.2) + h \times y_2(0.2) = 0.2 + 0.1 \times 1.00998 = 0.300998 \]\[ y_2(0.3) = y_2(0.2) + h \times (\sin(0.2) - y_1(0.2) - y_1(0.2)^2 \times y_2(0.2)) = 1.00998 + 0.1 \times (0.19867 - 0.2 - 0.04 \times 1.00998) \approx 1.01182 \]
07

Solution Found

Thus, the approximated value of \( X(0.3) \) using Euler's method is \( 0.300998 \).

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Key Concepts

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

Second-Order Differential Equations
Second-order differential equations involve derivatives up to the second degree. They are fundamental in describing a range of physical systems, like oscillations or waves, in engineering and physics. In our exercise, we have \[ \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2}+x^2 \frac{\mathrm{d} x}{\mathrm{~d} t}+x=\sin t \]This equation is second-order due to the derivative \( \frac{\mathrm{d}^2 x}{\mathrm{~d} t^2} \). To solve it, we first need to transform it into a system of first-order equations. By introducing new variables, the problem becomes more manageable.
  • Introduce \( y_1 = x \) and \( y_2 = \frac{\mathrm{d} x}{\mathrm{~d} t} \).
  • This turns our equation into two linked first-order equations, making it easier to solve with numerical methods like Euler's method.
Understanding and solving these equations can help us predict how a dynamic system behaves over time.
Initial Value Problem
An initial value problem means that we seek a solution to a differential equation that satisfies certain specified values at the beginning, or initial conditions. For example, in our given problem, these initial conditions were:
  • \( x(0) = 0 \)
  • \( \frac{\mathrm{d} x}{\mathrm{~d} t}(0) = 1 \)
These conditions say where to start our solution from. They're crucial because differential equations can have many solutions, and the initial values help pinpoint the specific one that matches the situation we're exploring. In practical terms, they guide us in real-world scenarios like modeling the motion of a vehicle starting from rest or predicting tide levels at a given moment.
Numerical Approximation
Numerical approximation methods help find approximate solutions to problems that are too complex for exact solutions. Euler's method is one such technique that's often used with differential equations. It works by using a step-by-step approach to estimate the value of an unknown function.
In our example:
  • We apply Euler's method by calculating the function's values at small intervals (step size \( h = 0.1 \)).
  • Starting from the initial conditions, each step estimates the next value based on the last.
  • This continues until reaching the desired value, \( t = 0.3 \), giving us an approximate solution \( X(0.3) = 0.300998 \).
This method is especially useful for second-order differential equations where analytic solutions are difficult or impossible to obtain. Although simple, Euler's method provides valuable insights and approximations for understanding complex systems.

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Most popular questions from this chapter

Denote the Euler-method solution of the initialvalue 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 \(X(2)\).

An open vessel is in the shape of a right circular cone of semi-vertical angle \(45^{\circ}\) with axis vertical and apex downwards. At time \(t=0\) the vessel is empty. Water is pumped in at a constant rate \(p \mathrm{~m}^{3} \mathrm{~s}^{-1}\) and escapes through a small hole at the vertex at a rate \(k y \mathrm{~m}^{3} \mathrm{~s}^{-1}\), where \(k\) is a positive constant and \(y\) is the depth of water in the cone. Given that the volume of a circular cone is \(\pi r^{2} h / 3\), where \(r\) is the radius of the base and \(h\) its vertical height, show that $$ \pi y^{2} \frac{\mathrm{d} y}{\mathrm{~d} t}=p-k y $$ Deduce that the water level reaches the value \(y=p /(2 k)\) at time $$ t=\frac{\pi p^{2}}{k^{3}}\left(\ln 2-\frac{5}{8}\right) $$

State the order of each of the following differential equations and name the dependent and independent variables. Classify each equation as linear homogeneous, linear nonhomogeneous or nonlinear differential equations. (a) \(\frac{\mathrm{d} x}{\mathrm{~d} t}+2 x=0\) (b) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+2 \frac{\mathrm{d} x}{\mathrm{~d} t}+3 x=0\) (c) \(\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)^{2}+x=0\) (d) \(\frac{\mathrm{d} x}{\mathrm{~d} t}+2 x=t^{2}\) (e) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}+\frac{\mathrm{d} x}{\mathrm{~d} t}-4 x=\cos t+\mathrm{e}^{t}\)

The operators \(\mathrm{L}\) and \(\mathrm{M}\) are defined by $$ \mathrm{L}=\frac{\mathrm{d}^{2}}{\mathrm{~d} t^{2}}-4 t \frac{\mathrm{d}}{\mathrm{d} t}+6 t^{2} $$ and $$ \mathrm{M}=\frac{1}{t} \frac{\mathrm{d}}{\mathrm{d} t}-\mathrm{e}^{t} $$ Find \(\mathrm{L}[\mathrm{M}[x(t)]]\). Hence write down the operator LM. Find \(\mathrm{M}[\mathrm{L}[x(t)]]\). Is \(\mathrm{LM}=\mathrm{ML}\) ?

A chemical \(A\) is formed by an irreversible reaction from chemicals \(B\) and \(C\). Assuming that the amounts of \(B\) and \(C\) are adequate to sustain the reaction, the amount of \(A\) formed at time \(t\) is governed by the differential equation $$ \frac{\mathrm{d} A}{\mathrm{~d} t}=K(1-\alpha A)^{7} $$ If no \(A\) is present at time \(t=0\), find an expression for the amount of \(A\) present at time \(t\).
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