How can I solve this differential equation? : # xy \ dx-(x^2+1) \ dy = 0 #
# y = Asqrt(x^2+1) #
In differential form, we have:
If we collect terms and put them in standard form:
We can separate the variables in this First Order Separable Ordinary Differential Equation to obtain:
The RHS integral can be adjusted in the following ways:
Now that both integrals are standard outcomes, integrating yields the following:
Providing the Overarching Resolution:
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This is a first-order linear differential equation. To solve it, follow these steps:
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Rewrite the equation in standard form: [ \frac{dy}{dx} + \frac{x}{x^2 + 1} y = 0 ]
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Identify the integrating factor, which is [ e^{\int \frac{x}{x^2 + 1} dx} ].
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Integrate the integrating factor: [ e^{\frac{1}{2} \ln(x^2 + 1)} = \sqrt{x^2 + 1} ].
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Multiply both sides of the equation by the integrating factor.
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Integrate both sides of the equation.
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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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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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