Understanding Linear Ordinary Differential Equations

An ordinary differential equation (ODE) is considered linear if it can be expressed in the general form:

$ a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = f(x) $

Key conditions for linearity are:

• The dependent variable ($y$) and all its derivatives appear only to the first power.
• There are no products involving $y$ or its derivatives (e.g., $y \cdot y'$, $y \cdot y''$).
• The coefficients ($a_i(x)$) depend only on the independent variable ($x$).

Analysis of ODE Options

1. Option 1: $(x + 1)\frac{dy}{dx} - y = e^x(x + 1)^2$

   This is a first-order ODE. It matches the linear form $a_1(x)\frac{dy}{dx} + a_0(x)y = f(x)$ with $a_1(x) = (x+1)$, $a_0(x) = -1$, and $f(x) = e^x(x+1)^2$. The variable $y$ and its derivative $\frac{dy}{dx}$ are to the first power, and coefficients depend only on $x$. Therefore, it is linear.

   Result: Linear

2. Option 2: $\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$

   This equation involves the inverse derivative $\frac{dx}{dy}$ and terms like $\frac{y}{x}$. It cannot be written in the standard linear form with respect to $y$. The presence of $\frac{dx}{dy}$ indicates non-linearity.

   Result: Non-linear

3. Option 3: $\frac{d^2y}{dx^2} + n^2x = 0$

   This is a second-order ODE. It can be represented as $1 \cdot \frac{d^2y}{dx^2} + 0 \cdot \frac{dy}{dx} + 0 \cdot y = -n^2x$. This fits the linear form $a_2(x)\frac{d^2y}{dx^2} + a_1(x)\frac{dy}{dx} + a_0(x)y = f(x)$ with $a_2(x)=1$, $a_1(x)=0$, $a_0(x)=0$, and $f(x)=-n^2x$. The coefficients are functions of $x$, and the derivatives are to the first power. Therefore, it is linear.

   Result: Linear

4. Option 4: $xy\frac{dy}{dx} = 1 + x + y + xy$

   This equation includes the term $y\frac{dy}{dx}$. This is a product of the dependent variable $y$ and its derivative $\frac{dy}{dx}$, which violates the condition for linearity.

   Result: Non-linear

Conclusion on Linear ODEs

The ordinary differential equations that satisfy the conditions for linearity are Option 1 and Option 3.