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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 15-28, find the general solution to the given differential equation.

26.   x 2 y d y d x = y 2 + 1

To determine

To calculate: The general solution to the differential equation x2ydydx=y2+1.

Explanation

Given Information:

The provided differential equation is x2ydydx=y2+1.

Formula used:

Solution of the differential equation g(y)dy=f(x)dx is g(y)dy=f(x)dx.

The logarithmic formula of integration is 1xdx=ln|x|+C.

The power of x formula of integration is xndx=xn+1n+1+C, where n1.

Convert the logarithmic equation lny=x into the exponential equation as y=ex.

The product rule for exponent is such that, aman=am+n, where a is a real number and m, n are integers.

Calculation:

Consider the differential equation, x2ydydx=y2+1

Rearrange the equation as,

yy2+1dy=1x2dx

Integrate both sides of the equation,

1y2+1ydy=1x2dx

If y2+1=t then,

2ydy=dtydy=12dt

Substitute y2+1=t and ydy=12dt in the equation 1y2+1ydy=1x2dx

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