Second-order with constant coefficients

Japanese school year: University year 2

What you learn

Learn how to solve second-order linear differential equations with constant coefficients using characteristic equations. They are fundamental in modeling oscillatory phenomena, such as spring vibrations and alternating current circuits. Prior knowledge of solving quadratic equations and differentiating exponential functions is recommended.

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Key points

This is the characteristic equation for solving a second-order linear ODE with constant coefficients. Assuming the solution $y = e^{\lambda x}$ turns the differential equation into a quadratic equation for $\lambda$ .

ay'' + by' + cy = 0 \implies a\lambda^2 + b\lambda + c = 0

This is the general solution when the characteristic equation has two distinct real roots $\lambda_1$ and $\lambda_2$ . It combines two exponential functions with arbitrary constants $C_1$ and $C_2$ .

y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x} \quad (\lambda_1 \neq \lambda_2)

This is the general solution when the characteristic equation has a repeated root $\lambda$ . The second term is multiplied by $x$ to provide a linearly independent solution ( $C_1, C_2$ are constants).

y = (C_1 + C_2 x)e^{\lambda x}

This is the general solution when the characteristic roots are complex numbers $\alpha \pm i\beta$ . It describes oscillation using $\cos\beta x$ and $\sin\beta x$ , with $e^{\alpha x}$ governing the amplitude ( $C_1, C_2$ are constants).

y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x) \quad (\lambda = \alpha \pm i\beta)

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Practice 1Coming soon