A second-order equation needs two constants

Find the general solution of the differential equation d2ydx2=sec⁡xtan⁡x\frac{d^2y}{dx^2}=\sec x\tan x. marks

[modified 2011/RVHS/Promo/12ai]

Solution

General solution

Integrating twice

Find the general solution of d2ydx2=sec⁡xtan⁡x\dfrac{d^2y}{dx^2}=\sec x\tan x.

Explore this idea

Hint

  1. Integrate once to get dydx\frac{dy}{dx}, with a constant.

  2. Integrate again with a second constant.

Answer · General solution

y=ln⁡∣sec⁡x+tan⁡x∣+Cx+Dy=\ln|\sec x+\tan x|+Cx+D

On intervals where cos⁡x≠0\cos x\ne0.

Explore the idea

D shifts the curve; C tilts it

Start with C = D = 0, then change each constant in turn.

Teal: C = 0, D = 0. No comparison overlay.

Start: C = 0, D = 0

y=ln⁡∣sec⁡x+tan⁡x∣y=\ln|\sec x+\tan x|: increasing, with an inflection at the origin.

Ideas to try
Which constant changes the value at x=0x=0? Which changes the gradient there?