Separable, linear and exact equations, integrating factors, constant coefficient second order equations and undetermined coefficients.
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- What is a separable differential equation?
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An equation where the variables can be separated to opposite sides, written as M(x)dx + N(y)dy = 0- Write the standard form of a separable equation
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M(x)dx + N(y)dy = 0- What is the key procedure to solve dy/dx = f(x)g(y)?
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Separate variables: dy/g(y) = f(x)dx, then integrate both sides- Solve dy/dx = y by separable method
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Separate: dy/y = dx, integrate: ln|y| = x + C, so y = Ae^x- Can y = 0 be a singular solution to a separable equation?
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Yes, if it satisfies the original equation but does not appear in the general solution for any value of C- For dy/dx = 2xy, find the solution by separating variables
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dy/y = 2x dx, integrate: ln|y| = x² + C, so y = Ae^(x²)- When solving a separable equation, what must you remember about constants of integration?
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You only need one arbitrary constant C on one side after integrating both sides- What form does the solution to a separable equation typically take?
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An implicit relation F(x) + G(y) = C, which may or may not be solvable for y explicitly- Write the standard form of a first-order linear differential equation
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dy/dx + P(x)y = Q(x)- What is an integrating factor?
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A function u(x) that, when multiplied by a linear equation, makes it exact (or a total derivative)- Give the formula for the integrating factor of dy/dx + P(x)y = Q(x)
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u(x) = e^(∫P(x)dx)- After multiplying dy/dx + P(x)y = Q(x) by integrating factor u(x), what does the left side become?
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d/dx[u(x)y], which is a total derivative- After using an integrating factor on dy/dx + P(x)y = Q(x), how do you solve for y?
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Integrate both sides: u(x)y = ∫u(x)Q(x)dx + C, then divide by u(x)- Solve dy/dx - 2y = 0 using an integrating factor
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Integrating factor: u(x) = e^(-2x). Then e^(-2x)y = C, so y = Ce^(2x)- What test determines if an equation M(x,y)dx + N(x,y)dy = 0 is exact?
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The equation is exact if and only if ∂M/∂y = ∂N/∂x
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