Standard form and the characteristic equation method

Key idea

For a homogeneous second-order linear ODE with constant coefficients, \[ y''+ay'+by=0, \] we try a solution of the form \(y=e^{rx}\). Then \(y'=re^{rx}\) and \(y''=r^2e^{rx}\). Substitute: \[ r^2e^{rx}+ar e^{rx}+b e^{rx}=0 \quad\Rightarrow\quad (r^2+ar+b)e^{rx}=0. \] Because e^{rx}≠ 0, we get the characteristic equation \[ r^2+ar+b=0. \] Solve this quadratic. The type of roots determines the correct general solution.

Root cases (memorize these templates)

• Two distinct real roots r_1≠ r_2: \(y=C_1e^{r_1x}+C_2e^{r_2x}\).
• Repeated real root \(r\): \(y=(C_1+C_2x)e^{rx}\).
• Complex roots \(r=\alpha\pm i\beta\): \(y=e^{\alpha x}\bigl(C_1\cos(\beta x)+C_2\sin(\beta x)\bigr)\).

Worked example

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Summary

• For \(y''+ay'+by=0\), solve \(r^2+ar+b=0\).
• Use the root templates to write the general solution quickly and correctly.

Two distinct real roots: factoring and fast solutions

Key idea

If the characteristic equation has two different real roots r_1≠ r_2, then \(e^{r_1x}\) and \(e^{r_2x}\) are linearly independent solutions, and the general solution is \[ y=C_1e^{r_1x}+C_2e^{r_2x}. \] Many quiz problems are designed so the quadratic factors nicely.

Worked example

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Summary

• Distinct real roots \(\Rightarrow\) sum of exponentials \(C_1e^{r_1x}+C_2e^{r_2x}\).
• If you’re given exponential solutions, read off the roots and build the characteristic polynomial.

Repeated roots: why the \(x e^{rx}\) term appears

Key idea

If the characteristic equation has a repeated root \(r\), you only get one exponential solution \(e^{rx}\) at first. To get a second linearly independent solution, you multiply by \(x\): \[ y_1=e^{rx},\qquad y_2=xe^{rx}. \] So the general solution is \[ y=(C_1+C_2x)e^{rx}. \]

Worked example

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Summary

• Repeated root \(r\Rightarrow y=(C_1+C_2x)e^{rx}\).
• Do not forget the \(x\) term — it’s what makes the second solution independent.

Complex roots: sinusoidal solutions and oscillation frequency

Key idea

If the characteristic equation has complex conjugate roots \[ r=\alpha\pm i\beta, \] then the real-valued general solution is \[ y=e^{\alpha x}\bigl(C_1\cos(\beta x)+C_2\sin(\beta x)\bigr). \] The parameter \(\beta\) is the angular frequency of oscillation. If \(\alpha=0\), the motion is a pure oscillation with constant amplitude. If \(\alpha<0\), the oscillations decay (damping). If \(\alpha>0\), the oscillations grow.

Worked example

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Summary

• Complex roots \(r=\alpha\pm i\beta\Rightarrow y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\).
• For \(y''+\beta^2y=0\), the oscillation frequency parameter is \(\beta\).

Quick connections: first-order linear ODEs and easy second-order reductions

Key idea

Some second-order linear ODEs are especially fast because the characteristic equation factors with \(r=0\) as a root. Examples include \(y''+y'=0\) and \(y''-4y'=0\). Treat them like any other constant-coefficient ODE: \[ y''+ay'+by=0\quad\Rightarrow\quad r^2+ar+b=0. \]

Worked example

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Summary

• Even “simple-looking” second-order linear ODEs are solved by the same characteristic equation method.
• First-order linear homogeneous \(y'+ky=0\Rightarrow y=Ce^{-kx}\).

Wronskian: checking linear independence of solutions

Key idea

For two differentiable functions \(y_1(x)\) and \(y_2(x)\), the Wronskian is \[ W(y_1,y_2)(x)= \begin{vmatrix} y_1 & y_2\\ y_1' & y_2' \end{vmatrix} = y_1y_2'-y_1'y_2. \] If W(y_1,y_2)(x_0)≠ 0 at some point \(x_0\), then \(y_1\) and \(y_2\) are linearly independent (and can be used to build the general solution \(y=C_1y_1+C_2y_2\)).

Worked example

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Summary

• \(W(y_1,y_2)(x)=y_1y_2'-y_1'y_2\).
• If W(x_0)≠ 0, the functions are linearly independent and can form the general solution.

Why second-order linear ODEs matter (and final practice set)

Where second-order linear ODEs show up

• Mechanical vibrations: mass-spring systems, resonance, oscillations (\(y''+\omega^2y=0\)).
• Electrical circuits: RLC circuits lead to \(y''+ay'+by=0\)-type models.
• Stability and damping: the sign of \(\alpha\) in \(e^{\alpha x}(\cdots)\) controls decay or growth.
• Modeling: many linearized systems near equilibrium reduce to constant-coefficient ODEs.

Worked example: another classic factorable case

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Final recap

• Homogeneous vs nonhomogeneous: \(y''+ay'+by=0\) vs \(y''+ay'+by=g(x)\).
• Characteristic equation: solve \(r^2+ar+b=0\) for constant-coefficient homogeneous equations.
• Distinct real roots: \(y=C_1e^{r_1x}+C_2e^{r_2x}\).
• Repeated root: \(y=(C_1+C_2x)e^{rx}\).
• Complex roots: \(y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\).
• Wronskian: W≠ 0 confirms linear independence of two solutions.