Question

In Exercises 31 and \(32,\) show that the function satisfies the differential equation. \(y=a \sinh x\) \(y^{\prime \prime \prime}-y^{\prime}=0\)

Short answer

The function \(y=a \sinh x\) satisfies the differential equation \(y^{\prime \prime \prime}-y^{\prime}=0\). The solution is verified by calculating the first, second and third derivatives of the function and substituting these back into the differential equation.

Step-by-step solution

Original Expression

We start with the original expression \(y=a \sinh x\).

First Derivative

We take the first derivative of this expression to get \(y' = a \cosh x\).

Second Derivative

Next, we calculate the second derivative. Differentiating again, we get \(y'' = a \sinh x\), which is the same as the original expression \(y\).

Third Derivative

Similarly, the third derivative is nothing but the first derivative, i.e., \(y''' = a \cosh x\).

Insert in Differential Equation

Now, we substitute all these calculated derivatives back into the differential equation. We get \(y''' - y' = a \cosh x - a \cosh x = 0\).

Evaluating the Expression

Solving the above, it is seen that the left-hand side equals the right hand side, which is zero. This shows that the function \(y=a \sinh x\) does indeed satisfy the differential equation \(y^{\prime \prime \prime}-y^{\prime}=0\).