Traps in Differential Equations
6 mistake patterns students fall for. 2 high-frequency traps appear in almost every exam.
Confusing order with degree
Order is the highest derivative present; degree is the exponent of that highest derivative when the DE is polynomial in derivatives.
Why: Both concepts relate to the highest derivative, so students mix them up. They may report the exponent as the order or vice versa.
Wrong integrating factor in linear DE
Computing the integrating factor incorrectly, often by using the wrong sign for P(x) or forgetting to rewrite the DE in standard form first.
Why: Students directly read off P(x) without bringing the DE to the standard form dy/dx + Py = Q, leading to a wrong IF.
Forgetting to divide by g(y) in variable separable
When separating variables in dy/dx = f(x)g(y), dividing by g(y) is valid only when g(y) is not zero. The case g(y) = 0 may give additional singular solutions.
Why: Students mechanically separate and integrate without checking if the denominator can be zero.
Not substituting back y = vx after solving homogeneous DE
After solving the separated equation in v and x, students forget to replace v with y/x to get the final answer.
Why: Students get the answer in terms of v and x and think they are done, forgetting that v was a temporary substitution.
Wrong sign in orthogonal trajectory substitution
When finding orthogonal trajectories, dy/dx must be replaced by -dx/dy (negative reciprocal), not just dx/dy or -dy/dx.
Why: Students forget the negative sign or invert incorrectly. Perpendicular slopes satisfy m1 * m2 = -1.
Missing constant of integration
Omitting the arbitrary constant C when writing the general solution of a DE, leading to an incomplete or incorrect answer.
Why: Students focus on the integration steps and forget that indefinite integration always produces a constant.
Can you spot these traps under time pressure?
Take a timed quiz on Differential Equations and see if you avoid the mistakes above.