Study guides › 9231 FP2 › Differential equations Maths4U Original · Learn, check, explain
Second order differential equations: The inhomogeneous case — 9231 FP2
Add a particular integral to the complementary function.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods For a linear constant-coefficient equation, the general solution is the complementary function plus one particular integral. An exponential forcing Keᶜˣ normally suggests \(\displaystyle Ae^{cx}\).
If c is an auxiliary root, multiply the trial by x for a simple root, or by \(\displaystyle x^{2}\) for a repeated root. Resonance concerns overlap with the complementary solution, not merely a large response. Determine remaining constants after adding the particular integral.
Check the conditions Check the forcing trial against the auxiliary roots. Substitute the trial to determine its coefficient. Three worked examples Example 1 · Inhomogeneous equations and resonance \(y''+6y'+8y=4e^{-2x},\ y(0)=2,\ y'(0)=3\). Write \(y=(A+Px)e^{-2x}+Be^{-4x}\).
Find P. Find A. Find B. Find y(1). Show the reasoning (a) \((4-2)P=4\implies P=2\)
Answer: \(2\).
(b) \(\displaystyle A+B=2,\ -2A+P-4B=3\implies A=\frac{9}{2}\)
Answer: \(\displaystyle \frac{9}{2}\).
(c) \(\displaystyle B=y(0)-A=- \frac{5}{2}\)
Answer: \(\displaystyle - \frac{5}{2}\).
(d) \(\displaystyle y=\left(2 x + \frac{9}{2}\right) e^{- 2 x} - \frac{5 e^{- 4 x}}{2}\); put \(\displaystyle x=1\).
Answer: \(0.833890243816147\).
Example 2 · Inhomogeneous equations and resonance \(y''+8y'+15y=5e^{-3x},\ y(0)=3,\ y'(0)=4\). Write \(y=(A+Px)e^{-3x}+Be^{-5x}\).
Find P. Find A. Find B. Find y(1). Show the reasoning (a) \(\displaystyle (5-3)P=5\implies P=\frac{5}{2}\)
Answer: \(\displaystyle \frac{5}{2}\).
(b) \(\displaystyle A+B=3,\ -3A+P-5B=4\implies A=\frac{33}{4}\)
Answer: \(\displaystyle \frac{33}{4}\).
(c) \(\displaystyle B=y(0)-A=- \frac{21}{4}\)
Answer: \(\displaystyle - \frac{21}{4}\).
(d) \(\displaystyle y=\left(\frac{5 x}{2} + \frac{33}{4}\right) e^{- 3 x} - \frac{21 e^{- 5 x}}{4}\); put \(\displaystyle x=1\).
Answer: \(0.499836763209339\).
Example 3 · Inhomogeneous equations and resonance \(y''+4y'+3y=6e^{-1x},\ y(0)=4,\ y'(0)=5\). Write \(y=(A+Px)e^{-1x}+Be^{-3x}\).
Find P. Find A. Find B. Find y(1). Show the reasoning (a) \((3-1)P=6\implies P=3\)
Answer: \(3\).
(b) \(A+B=4,\ -1A+P-3B=5\implies A=7\)
Answer: \(7\).
(c) \(B=y(0)-A=-3\)
Answer: \(-3\).
(d) \(y=\left(3 x + 7\right) e^{- x} - 3 e^{- 3 x}\); put \(\displaystyle x=1\).
Answer: \(3.52943320661083\).
Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
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