Recover independent solutions when roots repeat or leave the real line

Try

For y″+2y′+y=0, the characteristic polynomial is (r+1)². Writing C₁e^(−t)+C₂e^(−t) only produces one effective constant, C₁+C₂. It cannot fit arbitrary initial position and velocity. We need a genuinely independent second function, not a second name for the same mode.

Try y=te^(−t). Differentiate twice and substitute. The extra factor t changes the derivative enough to provide the missing freedom while still satisfying the equation.

Predict

If y=(C₁+C₂t)e^(−t), is y′(0) simply C₂? Predict the contribution from differentiating the exponential envelope. Then consider a complex root α+iβ: predict how the sign of α and the magnitude of β might affect the shape of a real-valued trajectory.

Experiment

Differentiate the repeated-root basis and verify it term by term.

Function	First derivative	Second derivative
e^(−t)	−e^(−t)	e^(−t)
te^(−t)	(1−t)e^(−t)	(t−2)e^(−t)

For the second row, the equation leaves [(t−2)+2(1−t)+t]e^(−t)=0. The Wronskian is e^(−2t), so the pair is independent. With y(0)=2 and y′(0)=1, C₁=2 and C₂−C₁=1, hence C₂=3. Check that the derivative of (2+3t)e^(−t) at zero is one.

Understand

A repeated real root r gives the basis e^(rt),te^(rt). This is a special case of a repeated factor in the differential operator, not permission to add t to every exponential trial indiscriminately. For a second-order constant-coefficient equation, these two functions provide the complete homogeneous family when the characteristic root is repeated.

For complex conjugate roots α±iβ with β nonzero, Euler’s identity e^(iθ)=cos θ+i sin θ yields a real basis e^(αt)cos βt and e^(αt)sin βt. Real coefficients and real initial data lead to real solutions, so complex roots do not imply a complex physical displacement. The exponential controls the envelope; β is angular frequency in radians per time unit.

For y″+4y′+13y=0, the roots are −2±3i. Write y=e^(−2t)(C cos3t+D sin3t). With y(0)=1, C=1. Differentiating both the envelope and the trigonometric factor gives y′(0)=−2C+3D. If the initial velocity is zero, D=2/3.

The resulting motion oscillates within an exponentially shrinking envelope. Its period in the oscillatory factor is 2π/3, but the full function is not periodic because the envelope changes from cycle to cycle. Calling every sine-containing solution periodic would conceal this distinction.

Check independence for the complex-root pair as well. Its Wronskian is βe^(2αt), which is nonzero when β is nonzero. The envelope derivatives cancel in that determinant. This supplies a reason that both initial conditions can be fitted, rather than merely observing two different-looking formulas. If β becomes zero, the sine basis collapses; at a repeated real root the time-multiplied exponential must replace it. Never substitute a zero frequency into a basis formula and assume the missing degree of freedom survives.

Apply

Draw a decaying envelope and sketch several oscillations inside it. Label the time between successive corresponding phases, and explain why their amplitudes differ. Compare with the repeated-root solution, which can rise initially before decaying without sustained oscillation.

Audit an incorrect initial-velocity calculation that ignores the envelope derivative. Identify the missing term and show how it changes D. Then verify one repeated-root and one complex-root solution directly. The characteristic equation chooses the basis; the initial data choose its coefficients, and both steps need checking.

Continue

Next add external forcing. When the forcing duplicates a homogeneous mode, the trial form must change again. The source of the extra time factor will be resonance rather than a repeated root alone.