Cooling coffee, charging circuits and vibrating machines all follow the same few equations. Learn to recognize them and the solutions follow.
Choose any section below to get a free practice problem. Work it with the FE Reference Handbook open, tap your answer, then check the solution.
If you can get every y on one side with dy and every x on the other with dx, integrate both sides and you are done.
dy/dt = −0.2y with y(0) = 100. What is y(5)?
y = 100e−0.2t. y(5) = 100e−1 = 36.8.
Common trap: (D) assumes 5 is a half-life; the half-life is ln 2/0.2 = 3.47.
Differential EquationsseparableThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Written as τ dy/dt + y = K u, a first-order linear system rises toward its steady value K u along 1 − e−t/τ.
What is the time constant of 3 dy/dt + 6y = 12?
Divide by 6: 0.5 dy/dt + y = 2. τ = 0.5 s.
Common trap: (D) reads the coefficient of dy/dt without dividing by the coefficient of y.
first-order lineartime constantintegrating factorThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
For ay″ + by′ + cy = 0, try y = erx to get the characteristic equation ar2 + br + c = 0.
What is the general solution of y″ + 4y′ + 4y = 0?
r2 + 4r + 4 = (r + 2)2: repeated root −2. y = (C1 + C2x)e−2x.
Common trap: (C) treats the double root as two different roots.
characteristic equationsecond-orderThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
A mass on a spring with a damper obeys m x″ + c x′ + k x = 0.
m = 1 kg, k = 100 N/m, c = 20 N·s/m. The system is:
ζ = 20/(2√100) = 20/20 = 1, so it is critically damped.
damping rationatural frequencyfree vibrationThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
The Laplace transform turns a differential equation into algebra: derivatives become multiplication by s.
What is the Laplace transform of f(t) = t?
From the table: ℒ{t} = 1/s2.
Common trap: (B) is ℒ{t2}.
Laplace Transformsfinal value theoremThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Newton’s Notes covers all 14 FE Mechanical exam areas: lessons, worked examples, hundreds of practice problems with full solutions, a 110-question practice exam, calculator keystrokes and Ask Newton, the AI tutor. Try it free for 3 days.
Start your 3-day free trialSee plans →