Nonlinear equation $6 + \sinh(x) = \sinh(3x)$

Question

I have this equation

• $6 + \sinh(x) = \sinh(3x)$

I know that I have to use this equation

• $\sinh(3x) = 3\sinh(x) + 4\sinh^3(x)$

• and substitution

Can anybody please help me? thx

Answer 1

Let $t=\sinh(x)$. Then $\sinh(3x)=4t^3+3t$. So your equation becomes $$ 6+t = 4t^3+3t, $$ $$ 4t^3+2t-6=0, $$ $$ (2t-2)(2t^2+2t+3)=0. $$ If a product of two factors is zero, then at least one of the factors must be zero: $$ 2t-2=0 \quad\mbox{ or }\quad 2t^2+2t+3=0. $$ The first factor gives us a real solution $$t=1, \quad \sinh x=1, \quad x=\sinh^{-1}(1)=\ln(1+\sqrt{2})\approx0.88137. $$ The second factor does not give us real solutions t (but there are complex ones).