How can I solve this differential equation? : # xy \ dx-(x^2+1) \ dy = 0 #

Answer 1

# y = Asqrt(x^2+1) #

In differential form, we have:

# xy \ dx-(x^2+1) \ dy = 0 #

If we collect terms and put them in standard form:

# 1/yy \ dy/dx = x/(x^2+1) #

We can separate the variables in this First Order Separable Ordinary Differential Equation to obtain:

# int \ 1/y \ dy = int \ x/(x^2+1) \ dx #

The RHS integral can be adjusted in the following ways:

# int \ 1/y \ dy = 1/2 \ int \ (2x)/(x^2+1) \ dx #

Now that both integrals are standard outcomes, integrating yields the following:

# ln|y| = 1/2ln|x^2+1| + C #
Noting that we require areal solution, and writing #C=lnA#, we get:
# ln y = ln Asqrt(x^2+1) #

Providing the Overarching Resolution:

# y = Asqrt(x^2+1) #
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Answer 2

This is a first-order linear differential equation. To solve it, follow these steps:

  1. Rewrite the equation in standard form: [ \frac{dy}{dx} + \frac{x}{x^2 + 1} y = 0 ]

  2. Identify the integrating factor, which is [ e^{\int \frac{x}{x^2 + 1} dx} ].

  3. Integrate the integrating factor: [ e^{\frac{1}{2} \ln(x^2 + 1)} = \sqrt{x^2 + 1} ].

  4. Multiply both sides of the equation by the integrating factor.

  5. Integrate both sides of the equation.

  6. Solve for y to get the general solution.

The general solution to the given differential equation is [ y = C(x^2 + 1)^{-\frac{1}{2}} ], where C is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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