Integration Involving Hyperbolic Functions

Question

Let y be a function of x.

Suppose we have, dy/dx = +/-[( (y^2)-c )/(sqrt(c))].

How do we integrate this to obtain, y = (sqrt(c))*{cosh[(x+D)/(sqrt(c))]} ?

Answer 1

integral= y(y^2-3c)/3sqr(c)+C

integral of (y^2-c)/sqrc dy

=y = (1/sqrc)integral of (y^2 -c)dy

= (1/sqrc) integral of y^2dy - (c)integral of 1dy

=(1/sqrc)(y^3/3) -cy + C

=y^3/3sqrc - 3ycsqrc/3sqrc + C

y= y(y^2 -3c)/3sqrc + C

if y= sqr(c)(cosh(x+D)/sqr(c)

then y=cosh(x+D) and

y'= sinh(x+D) which then must =dy/dx= (y^2-c)/sqr(c)

= (cosh^2(x+D)-c)/sqr(c)

= cosh^2(x+D)/sqr(c) - sqr(c)

=(1 - sinh^2(x+D))/sqr(c) - sqr(c)

= 1/sqr(c) - sqr(c) - sinh^2(x+D)/sqr(c)

= (1-c)/sqr(c) - sin^2(x+D)/sqr(c)

= (1-c-sin^2(x+D))/sqr(c)