Second-Order Linear Differential Equations

Precise definition

A second-order linear ODE has form $a(t)y''+b(t)y'+c(t)y=g(t)$ with $a(t)\ne0$ on an interval. For the homogeneous equation, two linearly independent solutions $y_1,y_2$ form a fundamental set and $y_h=C_1y_1+C_2y_2$. A nonhomogeneous solution is $y=y_h+y_p$.

Notation and mathematical language

The Wronskian $W=y_1y_2'-y_1'y_2$ tests independence under standard linear-equation hypotheses. Constant coefficients lead to a characteristic equation. Initial values $y(t_0),y'(t_0)$ determine $C_1,C_2$ uniquely when coefficients are continuous after normalization.

Conceptual picture

The homogeneous response represents free modes; forcing contributes a particular response. Changing $y_p$ by a homogeneous term does not change the full solution family, which is why any one particular solution suffices.

Conditions and key results

Superposition applies to the homogeneous operator and to forced responses with corresponding summed inputs, not to arbitrary nonlinear equations. Coefficient continuity and a nonzero leading coefficient define the regular interval. Boundary conditions at different points can yield zero, one, or many solutions.

A reliable strategy

1. Normalize the equation on a regular interval and solve the homogeneous problem.
2. Check independence and find a particular solution using a method suited to the forcing and coefficients.
3. Combine $y_h+y_p$ and apply both conditions without discarding derivative terms.
4. Verify the ODE, conditions, and interval; interpret homogeneous transients separately from forced behaviour.

Fully worked example

Interpretation and application

These equations model vibrations, circuits, beams in simplified settings, and feedback. Exact solution of a linear model does not quantify parameter uncertainty or nonlinear effects; those require separate validation.

Common mistakes

Verification and reasonableness

• Substitute $y,y',y''$ into the full nonhomogeneous equation.
• Evaluate the Wronskian or root structure for independence.
• Check both initial conditions and separate long-term limits from transient terms.

Practice

1. What is the homogeneous solution of $y''-y=0$?
2. What constant trial solves $y''+3y'+2y=6$?
3. How many independent homogeneous solutions are expected?

Further deduction

Abel's identity says the Wronskian for $y''+p(t)y'+q(t)y=0$ satisfies $W(t)=W(t_0)e^{-\int_{t_0}^t p(s)ds}$. Therefore, on a continuous-coefficient interval, a Wronskian nonzero at one point remains nonzero everywhere, while a zero value means dependence throughout. This is stronger than treating a single computed Wronskian value as an unexplained test.

For impulse-response thinking, choose the homogeneous solutions that satisfy $y_1(t_0)=1,y_1'(t_0)=0$ and $y_2(t_0)=0,y_2'(t_0)=1$. Then any unforced initial state is $y(t_0)y_1+y'(t_0)y_2$. A forcing response can be added through a Green-function integral. This separates initial-condition effects from later input and clarifies why exactly two independent data values control a regular second-order problem.

When one nonzero solution $y_1$ of $y''+P(x)y'+Q(x)y=0$ is known, reduction of order seeks $y_2=v(x)y_1(x)$. Substitution and cancellation reduce the problem to a first-order equation for $v'$, leading on a valid interval to $y_2=y_1\int e^{-\int P(x)dx}/y_1^2\,dx$. Zeros of $y_1$ require interval care. A nonzero Wronskian then confirms that the constructed solution adds a genuinely independent mode.

Related topics

• Characteristic Equation
• Undetermined Coefficients
• Variation of Parameters