/*! 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 8 Solve the differential equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 8

Solve the differential equation. $$ x y+y^{\prime}=100 x $$

Short Answer

Expert verified
The solution of the differential equation is: \(y = e^{-1/2x^2} * C + 200 e^{-1/2x^2}*\int{xe^{1/2x^2}} dx\), where \(C\) is an arbitrary constant.

Step by step solution

01

Write in General Form

First, rewrite this into linear differential equation standard form. The original equation is \(x y + y' = 100x\). Subtract \(x y\) from both sides to get \(y' + x y = 100 x\).
02

Find Integrating Factor

The integrating factor is found by calculating \(e^{\int P(x) dx}\), where \(P(x)\) is the coefficient in front of \(y\) in our equation, in this case \(P(x) = x\). So \(e^{\int x dx} = e ^{1/2x^2}\).
03

Multiply Original Equation

Multiply every term in the original equation by the calculated integrating factor: \(e^{1/2x^2}y' + x e^{1/2x^2}y =100x e^{1/2x^2}\).
04

Integration

This equation can now be written as the derivative of a product. So we integrate both sides of this equation: \(\int{d(e^{1/2x^2}y)} = \int{100x e^{1/2x^2} dx}\).
05

Calculate Result

Solving these integrals and rearranging gives the following solution for y: \(y = e^{-1/2x^2} * C + 200 e^{-1/2x^2}*\int{xe^{1/2x^2}} dx)\), where \(C\) is an arbitrary constant.

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

The chief financial officer of a company reports that profits for the past fiscal year were \(\$ 893,000\). The officer predicts that profits for the next 5 years will grow at a continuous annual rate somewhere between \(3 \frac{1}{2} \%\) and \(5 \%\). Estimate the cumulative difference in total profit over the 5 years based on the predicted range of growth rates.

If the portion of the line \(y=\frac{1}{2} x\) lying in the first quadrant is revolved about the \(x\) -axis, a cone is generated. Find the volume of the cone extending from \(x=0\) to \(x=6\).

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ g(x)=\frac{4 \ln x}{x}, \quad y=0, \quad x=5 $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=\sqrt{1+x^{3}}, \quad y=\frac{1}{2} x+2, \quad x=0 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.