2nd-Order Nonhomogeneous ODE | Homogeneous + Particular

📖 Tutorial｜Second-Order Nonhomogeneous ODE

1. Definition

A second-order constant-coefficient nonhomogeneous ODE has the form y″ + a y′ + b y = f(x). Its general solution is y_h + y*.

2. Symbols

Symbol	Meaning
y_h	general solution of the homogeneous equation
y*	one particular solution
C	constant RHS
b	coefficient of y

3. How this tool works

Solve the homogeneous equation y″+ay′+by=0 to get y_h;
For a constant RHS C, assume y*=A (constant);
Substitute: bA=C, so A=C/b (requires b≠0);
The full solution is y = y_h + C/b.

4. Steps

Enter a, b and the constant RHS C;
Click Calculate to see y_h and the particular solution;
Combine y = y_h + C/b.

5. Example

Example: y″+3y′+2y=4. y_h=C₁e^(−x)+C₂e^(−2x); y*=4/2=2; full solution y=C₁e^(−x)+C₂e^(−2x)+2.

6. Pitfalls

If b=0, a constant particular solution does not exist; try y*=Ax instead;
Pick just one particular solution, not a family;
This tool only handles a constant RHS, not e^x or sin x.

❓ FAQ｜Second-Order Nonhomogeneous ODE

What is the structure of the solution?

General solution = general solution of the homogeneous equation + one particular solution.

How to find a constant particular solution?

Set y*=A, substitute and compare constants: bA=C, so A=C/b.

What if b=0?

A constant particular solution does not exist; try y*=Ax.

What if the RHS is e^x or sin x?

Use undetermined coefficients with y* in the same form; this tool only handles a constant RHS.

Can the particular solution be chosen arbitrarily?

Any solution of the equation works; different choices differ by a homogeneous solution already captured by C₁, C₂.