Skip to main content
MathidiaMathematics Academy
Differential equations
Lesson 54 of 75
Challenge

Forced equations

Objective. Find complementary and particular solutions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Forced equations are inhomogeneous ODEs ay'' + by' + cy = f(x), solved as y = y_c + y_p, where y_c solves the homogeneous version and y_p is any particular solution matching the forcing f(x). Use undetermined coefficients: guess a particular form based on f(x), such as a constant for constant forcing or Asin(x)+Bcos(x) for sin(x) forcing, then substitute to find the coefficients. For y'' − y = 2, try y_p = k constant, giving −k = 2 so k = −2, combined with y_c = Ae^x + Be^(−x). If the guessed form already solves the homogeneous equation, multiply the guess by x.

Worked example

Find the general solution of the forced equation y'' − y = 6, a constant forcing term.

6
Coordinate grid with the line 6
  1. 1Homogeneous: λ²−1=0 gives y_c = Ae^x + Be^(−x)
  2. 2Try particular y_p = k constant
  3. 3Substitute: 0 − k = 6, so k = −6
  4. 4General solution: y = Ae^x + Be^(−x) − 6

Answer: y = Ae^x + Be^(−x) − 6

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Forgetting to add the complementary function y_c to the particular solution y_p, leaving the general solution with too few arbitrary constants to match the initial conditions given.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard