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Chapter 2: Problem 4

Find the Taylor series solution about \(x, y=0\) of the initial value problem \(u_{y}=\sin u_{x}, u(x, 0)=\pi x / 4\)

Short Answer

Expert verified
The Taylor series solution for the given initial value problem can be obtained by calculating the coefficients \(a_1, a_2, a_3, \ldots\), after differentiating the Taylor series and given partial derivative w.r.t. \(x\).

Step by step solution

01

Recognize the Initial Condition

Given in the problem is the initial condition \(u(x, 0) = \pi x / 4 \). This is the value of the function \(u(x, y)\) at \(y=0\), which is also the zeroth term in the Taylor series about \(y=0\).
02

Form the First Few Terms of the Taylor Series

A Taylor series of a function about a point can be written as \(f(x) = a_0 + a_1(x-a) + a_2(x-a)^2/2! + a_3(x-a)^3/3! + \ldots\). Here, \(f(x) = u(x, y)\), \(a = 0\), and the \(a_0, a_1, a_2, a_3, \ldots\) are coefficients to be determined. From the initial condition, we already know that \(a_0 = \pi x / 4\).
03

Differentiate the Taylor Series

We need to find the derivative of the function \(u(x, y)\) with respect to \(x\) to match the given partial derivative \(u_{y}=\sin u_{x}\). To do this, differentiate the Taylor series obtained in step 2 w.r.t. \(x\) using the power rule for differentiation.
04

Calculate Coefficients of the Taylor Series

Differentiate \(u_{y}=\sin u_{x}\) w.r.t. \(x\). Put these terms equal to the result obtained in step 3 to find the coefficients \(a_1, a_2, a_3, \ldots\). The result is the Taylor series representation of the function \(u(x, y)\).

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Most popular questions from this chapter

Find the Taylor series solution about \(x, y=0\) of the initial value problem \(u_{y}=\sin u_{x}, u(x, 0)=\pi x / 4\)

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. If \(a u_{x x}+2 b u_{x y}+c u_{y y}=d\) is elliptic (i.e., \(a c-b^{2}>0\) ), let \(W=\sqrt{a c-b^{2}}\). Show that solutions \(\mu, \eta\) of the Beltrami equations $$ \mu_{x}=\frac{b \eta_{x}+c \eta_{y}}{W} \quad \mu_{y}=-\frac{a \eta_{x}+b \eta_{y}}{W} $$ provide new coordinates transforming (7) to the form (14). [Note that \(\mu(x, y)=\int_{\gamma} \mu_{x} d x+\mu_{y} d y\), where \(\gamma\) is a path joining \((x, y)\) and a fixed point \(p_{0}\); path independence is provided by the Beltrami equations.]

Find the solution of the initial value problem \(u_{y y}=u_{x x}+u, u(x, 0)=e^{x}\), \(u_{y}(x, 0)=0\) in the form of power series expansion with respect to \(y\) [i.e., \(\left.\sum_{0}^{\infty} a_{n}(x) y^{n}\right]\). (Note: This is not a Taylor series.)
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