IntermediateTopic 64 of 104Engineering Mathematics
Undetermined coefficients
Guess the form of the particular solution based on the forcing term.
In depth
Guess y_p with the same form as the forcing: polynomial for polynomial, Ce^{at} for exponential, A cos+B sin for sinusoid. If the guess duplicates a homogeneous solution, multiply by t. Substitute and solve for the coefficients.
Key points
- Works for constant-coefficient ODEs
- Sums of forcing terms handled separately
- Multiply by t for resonance case
ExampleFor y''+y=e^{2x}, try y_p=Ae^{2x} giving A=1/5.
This method finds particular solutions quickly when the forcing is a polynomial, exponential or sinusoid, which covers most textbook and practical inputs.
What is Undetermined coefficients?
Guess the form of the particular solution based on the forcing term.
Why is it important to understand undetermined coefficients?
This method finds particular solutions quickly when the forcing is a polynomial, exponential or sinusoid, which covers most textbook and practical inputs.
When can I use undetermined coefficients?
Only when the forcing term is a combination of polynomials, exponentials and sines or cosines. Other forcing functions need variation of parameters.
What do I do if my guess duplicates a homogeneous solution?
Multiply the guess by t, or by t squared if needed, until it no longer duplicates any part of the complementary solution.
How is Undetermined coefficients used in practice?
Typical applications include finding the steady response of a spring to a sinusoidal force, solving a heat equation with a constant heat source and finding the particular solution of a circuit driven by a step input.
What should I learn before Undetermined coefficients?
Study "Homogeneous and particular solutions" first (topic 63 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Undetermined coefficients?
Works for constant-coefficient ODEs. Sums of forcing terms handled separately. Multiply by t for resonance case.
Can you give an example of Undetermined coefficients?
For y''+y=e^{2x}, try y_p=Ae^{2x} giving A=1/5.
What comes after Undetermined coefficients?
The next topic is "Variation of parameters", which builds directly on this one.
How difficult is Undetermined coefficients?
It is classed as Intermediate in Engineering Mathematics (topic 64 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.