/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 119 If \(y=e^{a x} \sin b x\), prove... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 119

If \(y=e^{a x} \sin b x\), prove that, \(y^{2}-2 a y_{1}+\left(a^{2}+b^{2}\right) y=0\)

Short Answer

Expert verified
After deriving the function twice, substituting \(y\), \(y'\), and \(y''\) into the equation produces a result of 0, proving the given equation.

Step by step solution

01

Differentiate the function

The first step involves differentiating \(y=e^{a x} \sin b x\). Keep in mind that this requires applying the product rule. The derivative of \(e^{ax}\) is \(ae^{ax}\), and the derivative of \(\sin b x\) is \(b \cos b x\). Therefore, \[y_1=y'= ae^{ax}sin(bx) + be^{ax}cos(bx)\].
02

Differentiate the function again

Now, differentiate \(y_1\) to obtain the second derivative \(y''\). It involves the product rule and the chain rule. The derivative of \(ae^{ax}\sin(bx)\) is \(a^2e^{ax}\sin(bx)+abe^{ax}\cos(bx)\), and the derivative of \(be^{ax}\cos(bx)\) is \(ab^2e^{ax}\sin(bx)-b^2e^{ax}\cos(bx)\). Thus, \[y_2=y'' = a^2e^{ax}\sin(bx)+2abe^{ax}\cos(bx)-b^2e^{ax}\cos(bx)\].
03

Substitute into the equation

Finally, substitute \(y\), \(y'\), and \(y''\) into the equation \(y''-2ay'+(a^2+b^2)y=0\). From steps 1 and 2, it is found that \(y'' - 2ay' + (a^2+b^2)y = a^2e^{ax}\sin(bx)+2abe^{ax}\cos(bx)-b^2e^{ax}\cos(bx) - 2a[ae^{ax}sin(bx) + be^{ax}cos(bx)] + (a^2+b^2)e^{ax}sin(bx) =0\]. This proves the given equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\begin{aligned} &\text { If } x=a(1-\cos \theta), y=a(\theta+\sin \theta) \text {, prove that }\\\ &\frac{d^{2} y}{d x^{2}}=-\frac{1}{a} \text { at } \theta=\frac{\pi}{2} . \end{aligned}

If \(y=e^{x}(\sin x+\cos x)\), prove that \(y_{2}-2 y_{1}+2 y=0\)

If \(y=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right), x>0\), then prove that \(\frac{d y}{d x}=\frac{4}{1+x^{2}}\).

Find \(\frac{d y}{d x}\), if \(y=x^{\sin x}\)

If \(y=\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)\), find \(\frac{d y}{d x} .\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.