Solution Found!
Consider the initial value problem (a) Using definite
Chapter 2, Problem 27E(choose chapter or problem)
Consider the initial value problem
\(\frac{d y}{d x}+\sqrt{1+\sin ^{2} x}\ y=x\), \(y(0)=2 \)
(a) Using definite integration, show that the integrating factor for the differential equation can be
written as
\(\mu(x)=\exp \left(\int_{0}^{x} \sqrt{1+\sin ^{2}t}\ d t\right)\)
and that the solution to the initial value problem is
\(y(x)=\frac{1}{\mu(x)}\int_0^x\mu(s)\ s\ ds+\frac{2}{\mu(x)}\).
(b) Obtain an approximation to the solution at \(x=1\) by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of \(\mu(x)\) and, thereby, the value of
\(\int_0^1\mu(s)\ s\ ds\).
[Hint: First, use Simpson’s rule to approximate at \(x=0.1\), 0.2, . . . , 1. Then use these values and apply Simpson’s rule again to approximate \(\int_0^1\mu(s)\ s\ ds\).]
(c) Use Euler’s method (Section 1.4) to approximate the solution at \(x=1\), with step sizes \(h=0.1\) and 0.05.
[A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]
Equation Transcription:
Text Transcription:
\frac{d y}{d x}+\sqrt{1+\sin ^{2} x}\ y=x
y(0)=2
\mu(x)=\exp \left(\int_{0}^{x} \sqrt{1+\sin ^{2}t}\ d t\right)
y(x)=\frac{1}{\mu(x)}\int_0^x\mu(s)\ s\ ds+\frac{2}{\mu(x)}
x=1
\mu(x)
\int_0^1\mu(s)\ s\ ds
x=0.1
\int_0^1\mu(s)\ s\ ds
x=1
h=0.1
Questions & Answers
QUESTION:
Consider the initial value problem
\(\frac{d y}{d x}+\sqrt{1+\sin ^{2} x}\ y=x\), \(y(0)=2 \)
(a) Using definite integration, show that the integrating factor for the differential equation can be
written as
\(\mu(x)=\exp \left(\int_{0}^{x} \sqrt{1+\sin ^{2}t}\ d t\right)\)
and that the solution to the initial value problem is
\(y(x)=\frac{1}{\mu(x)}\int_0^x\mu(s)\ s\ ds+\frac{2}{\mu(x)}\).
(b) Obtain an approximation to the solution at \(x=1\) by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of \(\mu(x)\) and, thereby, the value of
\(\int_0^1\mu(s)\ s\ ds\).
[Hint: First, use Simpson’s rule to approximate at \(x=0.1\), 0.2, . . . , 1. Then use these values and apply Simpson’s rule again to approximate \(\int_0^1\mu(s)\ s\ ds\).]
(c) Use Euler’s method (Section 1.4) to approximate the solution at \(x=1\), with step sizes \(h=0.1\) and 0.05.
[A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]
Equation Transcription:
Text Transcription:
\frac{d y}{d x}+\sqrt{1+\sin ^{2} x}\ y=x
y(0)=2
\mu(x)=\exp \left(\int_{0}^{x} \sqrt{1+\sin ^{2}t}\ d t\right)
y(x)=\frac{1}{\mu(x)}\int_0^x\mu(s)\ s\ ds+\frac{2}{\mu(x)}
x=1
\mu(x)
\int_0^1\mu(s)\ s\ ds
x=0.1
\int_0^1\mu(s)\ s\ ds
x=1
h=0.1
ANSWER:
Solution
Step 1
In this problem we have to that the integrating factor for the differential equation can be written as
