A second-order equation needs two constants · source reference

Printed question

Find the general solution of the differential equation $\frac{d^2y}{dx^2}=\sec x\tan x$. [2]

[modified 2011/RVHS/Promo/12ai]

Printed working

$y''=\sec x\tan x$.

$y'=\int\sec x\tan x\,dx=\sec x+c$.

$y=\int(\sec x+c)\,dx=\ln|\sec x+\tan x|+cx+d$, where $c,d$ are arbitrary constants.

Printed answer key

$y=\ln|\sec x+\tan x|+cx+d$, where $c,d$ are arbitrary constants.

Teacher note 1

Given DE of form $d^2y/dx^2=f(x)$. Integrate with respect to $x$ twice to obtain the general solution. There are 2 arbitrary constants in the general solution.

Mathematical review

Since $(\sec x)'=\sec x\tan x$, $y'=\sec x+C$. A further integral gives $y=\ln|\sec x+\tan x|+Cx+D$. Differentiating twice verifies the equation wherever $\cos x\ne0$. Constants are chosen separately on connected intervals between the poles.

Provenance

TJC JC2 2025 H2 Maths (9758) MSM. Native transcription checked against original visible question, working and sidebars. Question page(s): 1. Working page(s): 1. Original records remain private for QA; this native article has no PDF or raster worksheet dependency.