Review / Part I — Explicit & Qualitative Methods / Ch 1

Order, Linearity, and Solutions

Ch 1 — Part I — Explicit & Qualitative Methods

Two cheap-but-vital features of an ODE tell you most of what's possible: its order and whether it is linear.

Order

The order of an ODE is the order of the highest derivative of the unknown that appears. For $y''' + y'' - y = 0$, the order is 3.

Linear vs nonlinear

An ODE is linear in $y$ if it can be written $$a_n(t) y^{(n)} + a_{n-1}(t) y^{(n-1)} + \cdots + a_1(t) y' + a_0(t) y = g(t).$$ Here the coefficients $a_k(t)$ and forcing $g(t)$ can be arbitrary functions of $t$, but $y$ and its derivatives appear only to the first power and are not multiplied together. The equation is homogeneous if $g \equiv 0$.

Linear ODEs enjoy the superposition principle: any linear combination of homogeneous solutions is another homogeneous solution.

Quick tests

A product like $y y'$, a power like $y^2$, or $\sin(y)$ makes the equation nonlinear. A term like $t^2 y$ is fine — the coefficient is a function of the independent variable $t$, not of $y$.

Practice Problems

Problem 1easy

Give the order and determine linearity of $t^2 y'' - e^t y' + 3y = \sin t$.

Hint

Highest derivative → order. Are $y, y', y''$ appearing only to first power, not multiplied together?

Solution

Order 2. Coefficients depend only on $t$, and the unknown and its derivatives appear to the first power with no products. Linear and inhomogeneous.

Answer: Order 2, linear, inhomogeneous.

Problem 2easy

Classify $y' = y^2 - t$.

Hint

Spot the $y^2$ term.

Solution

Because of $y^2$, this is nonlinear. Order is 1.

Answer: Order 1, nonlinear.

Problem 3easy

Is $y'' + y y' = 0$ linear?

Hint

Look for products of the unknown with its derivatives.

Solution

The term $yy'$ multiplies the unknown by its derivative, violating linearity.

Answer: Nonlinear, order 2.

Problem 4medium

Write the general form of a linear third-order inhomogeneous ODE with $t$-dependent coefficients.

Hint

Use the template above with $n=3$.

Solution

$$a_3(t) y''' + a_2(t) y'' + a_1(t) y' + a_0(t) y = g(t).$$

Answer: See solution.

Problem 5easy

True or false: if $y_1$ and $y_2$ both solve the linear homogeneous equation $y'' + p(t) y' + q(t) y = 0$, then so does $c_1 y_1 + c_2 y_2$ for any constants.

Hint

Use linearity.

Solution

Substitute $y = c_1 y_1 + c_2 y_2$; derivatives distribute across the sum, and the $c_i$ factor out. Each $y_i$ contributes zero, so the whole sum is zero. This is superposition.

Answer: True.

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