Integrating factor & homogeneous equations
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Flip to reveal answersWhat is the integrating factor for dy/dx + P(x)y = Q(x)?
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All 8 Flashcards — Integrating factor & homogeneous equations
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Question
What is the integrating factor for dy/dx + P(x)y = Q(x)?
Answer
I = e^(∫P dx). Multiply through by it; the left side becomes (Iy)′.
Question
After multiplying a linear ODE by its integrating factor I, what is the left side?
Answer
Exactly (I·y)′ — so integrate to get Iy = ∫IQ dx.
Question
How do you spot P(x) in dy/dx + P(x)y = Q(x)?
Answer
Write the equation with dy/dx alone (coefficient 1); P(x) is then the coefficient of y.
Question
When is a first-order ODE homogeneous?
Answer
When dy/dx can be written using only the ratio y/x (e.g. (x+y)/x = 1 + y/x).
Question
What substitution solves a homogeneous ODE, and what is dy/dx then?
Answer
Let y = vx; then dy/dx = v + x dv/dx (product rule). It becomes separable in v and x.
Question
After solving in v, what's the final step?
Answer
Replace v by y/x, then apply the initial condition.
Question
∫cot x dx = ?
Answer
ln|sin x| + C (so e^(∫cot x dx) = sin x).
Question
Integrating factor for dy/dx + (2/x)y = x (x > 0)?
Answer
∫(2/x)dx = ln x², so I = x²; then (x²y)′ = x³.
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Differential equations (HL only)
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