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Exact Differential Equation Calculator

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Example
Created on 2024-06-20Asked by Benjamin Roberts (Solvelet student)
Solve the exact differential equation (2xy+1)dx+(x2+2y)dy=0 (2xy + 1)dx + (x^2 + 2y)dy = 0 .

Solution

To solve the exact differential equation (2xy+1)dx+(x2+2y)dy=0 (2xy + 1)dx + (x^2 + 2y)dy = 0 :Step 1: Check if the equation is exact by verifying if ∂M∂y=∂N∂x \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} , where M=2xy+1 M = 2xy + 1 and N=x2+2y N = x^2 + 2y : ∂M∂y=2xand∂N∂x=2x, \frac{\partial M}{\partial y} = 2x \quad \text{and} \quad \frac{\partial N}{\partial x} = 2x, so the equation is exact. Step 2: Find the integrating factor μ(x) \mu(x) by solving the equation ∂μ∂x=∂M∂y−∂N∂xN \frac{\partial \mu}{\partial x} = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} : ∂μ∂x=2x−2xx2+2y=0  ⟹  μ(x)=e0=1. \frac{\partial \mu}{\partial x} = \frac{2x - 2x}{x^2 + 2y} = 0 \implies \mu(x) = e^{0} = 1. Step 3: Multiply both sides of the equation by the integrating factor μ(x)=1 \mu(x) = 1 : (2xy+1)dx+(x2+2y)dy=0. (2xy + 1)dx + (x^2 + 2y)dy = 0. Step 4: Integrate both sides with respect to x x to find the general solution. Step 5: Integrate both sides with respect to y y to find the general solution. Step 6: Write the general solution by combining the results of steps 4 and 5. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Emma Rivera on Solvelet
1. Determine whether the differential equation (2xy+y2) dx+(x2+2xy) dy=0 (2xy + y^2) \, dx + (x^2 + 2xy) \, dy = 0 is exact.2. Solve the exact differential equation (2x+y) dx+(x+2y) dy=0 (2x + y) \, dx + (x + 2y) \, dy = 0 .
DefinitionAn exact differential equation is an ordinary differential equation (ODE) for which the substituted form M(x,y)dx + N(x,y)dy = 0 is exact (meaning that it is a total differential). An exact differential equation has integrating factors so that it can be derived by the total differential of a function and can be integrated directly. In the physical sciences, exact differential equations are used to model a number of different activities, such as heat flow, flow of fluids, population growth, among otherthings. For Example,The given exact differential equation(2xy+y2)dx+(x2+2xy)dy=0 can be easily recognized as exact as here ∂y∂​(2xy+y2)=∂x∂​(x2+2xy), so we can rewrite this form as d(x2y+31​y3)=0
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