For Each Equation Determine Whether It Is Linear

You've been staring at a list of equations for twenty minutes, pencil hovering over your notebook, and you just want to know: which ones are linear? Practically speaking, the good news? It's a question that shows up in differential equations, linear algebra, and even when you're checking if a system behaves nicely. In practice, which ones aren't? Once you know what to look for, it becomes second nature.

What Does "Linear" Actually Mean?

Before we start checking equations, let's get clear on what we mean by "linear." In the context of equations—especially differential equations—linear doesn't mean a straight line (though that's part of it). It means the equation can be written in a specific form where the dependent variable and its derivatives appear only to the first power and aren't multiplied together.

Here's the key: an equation is linear if it looks like this:

L(y) = f(x)

Where L is a linear operator—which basically means it's a combination of differentiation, multiplication by functions of x, and addition. The dependent variable y and its derivatives y', y'', y''', etc. can't be squared, cubed, multiplied by each other, or appear inside functions like sine or exponential.

Why This Matters

You might be thinking, "Okay, but why do I care if an equation is linear?They obey superposition: if y₁ and y₂ are solutions, then any combination c₁y₁ + c₂y₂ is also a solution. Linear equations are predictable in a way nonlinear ones aren't. Practically speaking, " Turns out, it matters a lot. This property makes linear equations much easier to solve and analyze.

In physics, engineering, and economics, linear models are often the starting point because they're tractable. Now, when you can't solve the full nonlinear version, you linearize it around an operating point and work with that approximation. So knowing how to spot linearity is practically useful, not just academically.

How to Check If an Equation Is Linear

Here's the systematic approach I use. For each equation, ask yourself these questions:

Is the dependent variable to the first power only?

If you see y², y³, √y, or 1/y, it's nonlinear. If y appears as y (to the first power), that's good.

Are derivatives to the first power only?

Same rule applies to y', y'', y''', etc. If you see (y')², sin(y'), or e^(y''), it's nonlinear.

Are the dependent variable and its derivatives multiplied together?

If you see terms like yy', y'y'', or y''y''', it's nonlinear. Each term should involve only one instance of y or its derivatives.

Are the dependent variable or derivatives inside nonlinear functions?

If you see sin(y), ln(y'), or e^(y''), it's nonlinear. Trigonometric, exponential, logarithmic functions of y or its derivatives break linearity.

Let's put this into practice with some examples.

Walking Through Examples

Example 1: y'' + 3y' + 2y = 0

This is a classic linear homogeneous equation. The coefficients (3 and 2) are just constants multiplying the derivatives. Plus, no products, no nonlinear functions. That said, each term involves y or its derivatives to the first power. Linear.

Example 2: y' + y² = 0

Here we have y², which is y to the second power. Because of that, that's enough to make this nonlinear. Even though the y' term is fine, that y² kills it.

Example 3: yy' + y = 5

Look at that first term: yy'. That's y multiplied by y'. Products of the dependent variable and its derivatives make equations nonlinear. This one's out.

Example 4: y'' + sin(x)y = 0

This might look tricky at first glance. But notice: sin(x) is a function of x only, not y. No products of y terms, no nonlinear functions of y. Because of that, the dependent variable y appears to the first power, and y'' is a second derivative to the first power. It's multiplying y, which is allowed in linear equations. This is linear.

Example 5: y' = e^y

Here we have e^y, which is an exponential function of the dependent variable. That's a clear violation of linearity. Even though we could rewrite this as y' - e^y = 0, that e^y term makes the whole thing nonlinear.

Example 6: (y')² + y = 0

That (y')² is y' squared. Practically speaking, derivatives can only appear to the first power in linear equations. This one's nonlinear, period.

Example 7: y''' + xy' + y = x²

Let's check each term: y''' is fine (first power), xy' is x times y' (allowed—x is the independent variable), y is fine, and x² is just a function of x on the right side. Everything checks out. Linear.

Example 8: y + y'y'' = 1

That middle term y'y'' is a product of two derivatives. This leads to that's not allowed in linear equations. Nonlinear.

Common Patterns That Break Linearity

After working through dozens of equations, certain patterns keep showing up as nonlinear offenders:

Powers higher than one: Any time you see yⁿ where n ≠ 1, or (y')ⁿ where n ≠ 1, you're dealing with a nonlinear equation.

Products of y terms: yy', yy'', y'y'', and so on. These products create coupling that linear equations don't have.

Nonlinear functions of y: This includes trigonometric functions (sin, cos, tan), exponential functions (e^y), logarithmic functions (ln y), and any other nonlinear transformation applied to y or its derivatives.

Functions of y in denominators: 1/y, 1/y', etc. These are equivalent to y⁻¹, which is nonlinear.

What About Systems of Equations?

Sometimes you'll have multiple equations with multiple dependent variables. So the same rules apply, but now you need to check each equation. A system is linear if every equation in it is linear.

For example:

Equation 1: y₁' + y₂ = 0
Equation 2: y₁ + y₂' = x

Both equations are linear (no powers, products, or nonlinear functions), so the system is linear.

But if equation 2 were y₁y₂' + y₂ = x, then the system would be nonlinear because of that product term.

The Short Version

Here's my quick checklist for determining linearity:

Check the powers: y, y', y'' all to the first power only? Good. Any squares, cubes, or other powers? Nonlinear.
Look for products: Any multiplication between y terms or between y and its derivatives? Nonlinear.
Scan for nonlinear functions: sin(y), e^(y'), ln(y'')? Nonlinear.
Remember the coefficients: Functions of x multiplying y terms are fine. Functions of y are not.

If you pass all four checks, you've got a linear equation.

Real Talk About the Tricky Cases

Some equations look like they might be nonlinear but actually are linear. And take y'' + p(x)y' + q(x)y = g(x). Consider this: this is the general form of a second-order linear differential equation. The coefficients p(x) and q(x) can be any functions of x—they don't have to be constants. As long as they don't involve y or its derivatives, you're good.

On the flip side, some equations that look linear aren't. Here's the thing — i've seen students miss that yy' is a product and call the equation linear. Or they'll see e^x and think "exponential, so nonlinear," forgetting that e^x as a coefficient multiplying y is perfectly fine.

Practical Tips for Getting This Right

Rewrite the equation first: Sometimes equations aren't obviously linear or nonlinear until you manipulate them. If you can isolate everything involving y and its derivatives on one side, that often makes it clearer.

For more on this topic, read our article on how to find the total resistance in a parallel circuit or check out unit 6 similar triangles homework 2 similar figures answer key.

Focus on what's multiplied together: The biggest giveaway is usually products. Train your eye to spot when y terms are multiplied by each other.

Don't overthink coefficients: Functions of x multiplying y terms are coefficients, not part of the y term itself. They don't affect linearity.

Use algebraic manipulation:

Solving Linear Differential Equations — What to Expect

Once you’ve confirmed that an equation is linear, the next step is to solve it. The good news is that linear equations enjoy a powerful toolbox that makes them far more tractable than their nonlinear cousins. Below are the most common techniques, illustrated with the simplest representative forms.

1. First‑Order Linear ODEs

A first‑order linear equation can always be written as

[ y' + p(x),y = q(x), ]

where (p) and (q) are arbitrary functions of the independent variable (x). The standard method is to multiply by an integrating factor

[ \mu(x)=e^{\int p(x),dx}, ]

which transforms the left‑hand side into the derivative of a product:

[ \frac{d}{dx}!\bigl[\mu(x),y\bigr]=\mu(x),q(x). ]

Integrating both sides with respect to (x) yields

[ \mu(x),y = \int \mu(x),q(x),dx + C, ]

and finally

[ y(x)=\frac{1}{\mu(x)}\Bigl(\int \mu(x),q(x),dx + C\Bigr). ]

Example.* Solve (y' - 2y = e^{x}).
Here (p(x)=-2), so (\mu(x)=e^{-2x}).

[ \frac{d}{dx}!\bigl[e^{-2x}y\bigr]=e^{-2x}e^{x}=e^{-x}. ]

Integrating,

[ e^{-2x}y = -\int e^{-x},dx + C = e^{-x}+C, ]

hence

[ y = e^{2x}\bigl(e^{-x}+C\bigr)=e^{x}+Ce^{2x}. ]

2. Constant‑Coefficient Homogeneous Equations

When the coefficients are constants, the equation reduces to algebraic characteristic equations. For an (n)‑th order homogeneous linear ODE

[ a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y' + a_0 y = 0, ]

assume a trial solution (y=e^{rx}). Substituting yields the characteristic polynomial

[ a_n r^{,n}+a_{n-1} r^{,n-1}+ \dots + a_1 r + a_0 = 0. ]

The roots (r_1,\dots,r_k) dictate the basis of solutions:

Distinct real roots → (e^{r_i x}).
Repeated real roots → (x^m e^{r x}) for multiplicity (m+1).
Complex conjugate pair ( \alpha \pm i\beta) → (e^{\alpha x}\cos\beta x) and (e^{\alpha x}\sin\beta x).

Example.In practice, * Solve (y''-3y'+2y=0). The characteristic equation is (r^2-3r+2=0) with roots (r=1,2).

[ y(x)=C_1 e^{x}+C_2 e^{2x}. ]

3. Non‑Homogeneous Linear Equations

If a forcing term (g(x)) appears on the right‑hand side, the equation is non‑homogeneous:

[ L[y]=g(x), ]

where (L) denotes the linear differential operator. The general solution is the sum of two parts:

[ y_{\text{general}}(x)=y_h(x)+y_p(x), ]

where (y_h) solves the associated homogeneous equation and (y_p) is any particular solution of the full equation.

Two workhorse methods for constructing (y_p) are:

Method of Undetermined Coefficients – works when (g(x)) is a linear combination of exponentials, sines, cosines, polynomials, or products thereof. One guesses a trial form with undetermined coefficients, substitutes, and solves for them.
Variation of Parameters – universally applicable. Using a fundamental set ({y_1,\dots,y_n}) of solutions to the homogeneous problem, one writes

[ y_p = \sum_{i=1}^{n} u_i(x) y_i(x), ]

where the functions (u_i) satisfy a linear system derived from imposing

[ \sum_{i=1}^{n} u_i'(x) y_i(x)=0. ]

Solving this system yields the (u_i) and thus (y_p).

Example (Variation of Parameters).*
Solve (y''+y = \tan x).
Homogeneous solutions: (y_1=\cos x,; y_2=\sin x).

[ y_p = u_1(x)\cos x + u_2(x)\sin x, ]

with the constraints (u_1'\cos x + u_2'\sin x = 0) and ( -u_1'\sin x + u_2'\cos x = \tan x).
Solving yields

[ u_

Solving the system

[ u_1'\cos x + u_2'\sin x = 0,\qquad -u_1'\sin x + u_2'\cos x = \tan x, ]

we obtain

[ u_1'=-\sin x\tan x=-\frac{\sin^2 x}{\cos x},\qquad u_2'=\cos x\tan x=\sin x. ]

Integrating,

[ u_1=\int-\frac{\sin^2 x}{\cos x},dx =\int\left(\frac{1}{\cos x}-\cos x\right)dx =\ln!\bigl|\sec x+\tan x\bigr|-\sin x, ]

[ u_2=\int\sin x,dx=-\cos x. ]

Thus

[ y_p = u_1\cos x+u_2\sin x = \cos x!\left[\ln!\bigl|\sec x+\tan x\bigr|-\sin x\right] -\cos x\sin x, ]

[ y_p=\cos x,\ln!\bigl|\sec x+\tan x\bigr|-\sin x\cos x. ]

The general solution is therefore

[ y(x)=C_1\cos x+C_2\sin x +\cos x,\ln!\bigl|\sec x+\tan x\bigr| -\sin x\cos x. ]

4. Systems of Linear Differential Equations

Many problems involve several unknown functions coupled together. Writing such a system in vector form,

[ \mathbf{Y}'(x)=A(x),\mathbf{Y}(x)+\mathbf{G}(x), ]

reduces the analysis to matrix exponentials when (A) is constant. If (A) has eigenvalues (\lambda_i) with corresponding eigenvectors (\mathbf{v}_i), then each eigenvalue–eigenvector pair contributes a solution

[ \mathbf{Y}_i=e^{\lambda_i x}\mathbf{v}_i. ]

When eigenvalues are complex, Euler’s formula converts the complex exponentials into real sinusoidal terms, mirroring the scalar case discussed earlier.

5. Series Solutions

For equations whose coefficients are not constant—particularly those arising in physics such as Bessel’s equation or Legendre’s equation—power-series methods are indispensable. Assuming a solution of the form

[ y(x)=\sum_{n=0}^{\infty}a_n(x-x_0)^n, ]

substituting into the differential equation, and equating coefficients of like powers yields a recurrence relation for the (a_n). This approach often leads to well-known special functions and provides a systematic way to handle singular points as well.

Conclusion

Linear differential equations form a cornerstone of applied mathematics, offering a rich interplay between algebraic structure and analytic techniques. Whether through integrating factors for first-order equations, characteristic polynomials for constant-coefficient problems, or variation of parameters and series expansions for more general cases, each method contributes to a versatile toolkit. Mastery of these foundational strategies equips one to tackle everything from simple exponential growth models to complex systems governed by partial differential equations, making them essential for both theoretical exploration and practical problem-solving across science and engineering.

For Each Equation Determine Whether It Is Linear