Question

The linear differential equation \[ a_0\frac{d^ny}{dx^n}+a_1\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_ny=F(x) \] is non-homogeneous if:

• A. \(F(x)=0\)
• B. \(F(x)\neq 0\)
• C. Degree is greater than order
• D. Order is greater than degree

Hint:

For a linear differential equation, right-hand side \(0\) means homogeneous and right-hand side non-zero means non-homogeneous.

Answer 1

The Correct Option is B

Concept:
A linear differential equation is called homogeneous when the right-hand side is zero. \[ a_0\frac{d^ny}{dx^n}+a_1\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_ny=0 \]

Step 1: Identify the right-hand side.
The given equation is: \[ a_0\frac{d^ny}{dx^n}+a_1\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_ny=F(x) \] Here, the right-hand side is: \[ F(x) \]

Step 2: Condition for non-homogeneous equation.
If: \[ F(x)\neq 0 \] then the differential equation is called non-homogeneous.

Step 3: Final conclusion.
Therefore, the equation is non-homogeneous when: \[ F(x)\neq 0 \] \[ \therefore \text{Correct Answer is (B)} \]