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Solve the IVP (x \ln (y^4) + 4 \ln y - y^2) dx + \left ( {-2y + \frac{4x}{y}} \right ) dy = 0, ...

Question:

Solve the IVP

{eq}(x \ln (y^4) + 4 \ln y - y^2) dx + \left ( {-2y + \frac{4x}{y}} \right ) dy = 0, \; y(0) = 1 {/eq}.

Solving a Nonexact Differential Equation:

In order to solve the differential equation {eq}M(x, y) \: dx + N(x, y) \: dy = 0, {/eq} we first check to see if it is exact. An exact differential equation satisfies the condition {eq}\displaystyle\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}. {/eq} If the condition is not satisfied but {eq}\displaystyle\frac{M_y - N_x}{N} {/eq} is a function of {eq}x {/eq} alone, then the function {eq}\mu(x) = e^{\int \frac{M_y - N_x}{N}} \: dx {/eq} is an integrating factor for the differential equation. After multiplying the equation by the integrating factor, the new equation will be exact.

Answer and Explanation: 1

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Given the initial value problem

{eq}(x \ln (y^4) + 4 \ln y - y^2) dx + \left ( {-2y + \displaystyle\frac{4x}{y}} \right ) dy = 0, \; y(0) = 1 {/eq},...

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Nonexact Equations: Integrating Factors

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Chapter 16 / Lesson 2
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The integrating factor method is useful in solving non-exact, linear, first-order, partial differential equations. Learn the technique of the integrating factors method and its application to the Fundamental Theorem of Calculus.


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