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Solve the following IVP's. (a) (2 x y^3 + cos y) dx + (-x sin y + 3 x^2 y^2) dy = 0, y (pi / 2) =...

Question:

Solve the following IVP's.

(a) {eq}\displaystyle (2 x y^3 + \cos y)\ dx + (-x \sin y + 3 x^2 y^2)\ dy = 0,\ y \bigg(\dfrac \pi 2\bigg) = 2 {/eq}.

(b) {eq}\displaystyle (x \ln (y^4) + 4 \ln y - y^2)\ dx + \bigg(-2 y + \dfrac {4 x} y\bigg)\ dy = 0,\ y(0) = 1 {/eq}.

Exact Differential Equation:

An equation with derivative term is known as differential equation.

Equation of form M(x,y)dx+N(x,y)dy=0 is known as exact differential equation when it satisfies the condition {eq}\displaystyle \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} {/eq},

Otherwise for non exact differential equation find integrating factor {eq}\displaystyle I.F=e^{\int \:\mu \left(x\right)dx\:}\\ \displaystyle where,\:\mu \:\left(x\right)=\frac{\frac{\partial \:M}{\partial \:y}-\frac{\partial \:N}{\partial \:x}}{N} {/eq} and multiply it with original equation to find solution.

Solution will be

{eq}\displaystyle \int _{y\:consatnt}M\:dx+\int _{No\:x\:term}N\:dy=C\:\: {/eq}

Use initial condition to find the value of constant.

Answer and Explanation: 1

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Given ODE is

(a) {eq}\displaystyle (2 x y^3 + \cos y)\ dx + (-x \sin y + 3 x^2 y^2)\ dy = 0,\ y \bigg(\dfrac \pi 2\bigg) = 2\\ {/eq}

By comparing...

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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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