 | Term | Definition |
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Homogeneous Differential Equation |
an equation of the form M(x,y)dx + N(x,y)dy = 0 where M and N are homogeneous functions of the same degree |
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Change of Variables for homogeneous Equations |
If M(x,y)dx + N(x,y)dy = 0 is homogeneous, then it can be transformed into a differential equation whose variables are separable by the substitution y = vx where v is a differentiable function of x. |
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Exact differential equation |
∂ƒ/∂x = M(x,y) and ∂ƒ/∂y = N(x,y) where M(x,y)dx + N(x,y)dy = 0 |
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Test for exactness |
∂M/∂x = ∂N/∂y |
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Integrating factor |
Consider the differential equation M(x,y)dx + N(x,y)dy =0 1. If [∂M/∂y - ∂N/∂x]/N = h(x) then e^∫h(x)dx is an integrating factor. 2. If [∂N/∂x - ∂M/∂y]/M = k(y) then e^∫k(y)dy is an integrating factor |
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First Order Linear Differential Equation |
dy/dx + P(x)y = Q(x) |
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Solution of a First-Order Linear Differential Equation |
ye^∫P(x)dx = ∫Q(x)e^∫P(x)dx dx + C |
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Bernoulli equation |
y' + P(x)y = Q(x)yⁿ |
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Solution to Bernoulli equation |
y¹⁻ⁿe^∫(1-n)P(x)dx = ∫(1-n)Q(x)e^∫(1-n)P(x)dx dx + C |
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Separable variable equation |
M(x)dx + N(y)dy = 0 |
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Linear Differntial Equation of order n |
Dⁿy + g₁(x)Dⁿ⁻¹y.... = f(x) |
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