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Chapter 20: Problem 5

Find a second-order differential equation that is satisfied by $$ y=A \cosh 2 x+B \sinh 2 x $$

Short Answer

Expert verified
Answer: The second-order differential equation satisfied by the given function is \(\frac{d^2y}{dx^2} = 4y\).

Step by step solution

01

Find the first derivative of the function

To find the first derivative of the function with respect to x, we apply the power and chain rule: $$ \frac{dy}{dx} = \frac{d}{dx}(A \cosh 2x + B \sinh 2x) = 2A \sinh 2x + 2B \cosh 2x $$
02

Find the second derivative of the function

Now, we find the second derivative of the function with respect to x by differentiating the first derivative: $$ \frac{d^2y}{dx^2} = \frac{d}{dx} (2A \sinh 2x + 2B \cosh 2x) = 4A \cosh 2x + 4B \sinh 2x $$
03

Find a relationship between the function and its derivatives

We can rewrite the second derivative by factoring out a 4 from both terms: $$ \frac{d^2y}{dx^2} = 4(A \cosh 2x + B \sinh 2x) $$ Notice that the expression inside the parentheses is the same as the original function, \(y\). Therefore, we have found a relationship between the original function and its second derivative: $$ \frac{d^2y}{dx^2} = 4y $$
04

Write the final answer

The second-order differential equation satisfied by the given function is $$ \frac{d^2y}{dx^2} = 4y $$

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Most popular questions from this chapter

Using software, obtain a symbolic solution of \(L \frac{\mathrm{d} i}{\mathrm{~d} t}+R i=10\left(1-\mathrm{e}^{-0.1 t}\right)\) when \(i(0)=i_{0}\)

Obtain the general solutions, that is the complementary functions, of the following homogeneous equations: (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+y=0\) (b) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} t}+5 y=0\) (c) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}-2 y=0\) (d) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+9 y=0\) (e) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) (f) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-16 x=0\)

By integrating twice find the general solution of \(y^{\prime \prime}=12 x^{2}\)

If \(y=A \mathrm{e}^{2 x}\) is the general solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2 y\), find the particular solution satisfying \(y(0)=3\). What is the particular solution satisfying \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2\) when \(x=0 ?\)

The equation governing the flow of current \(i\) in a series \(L R\) circuit with applied constant voltage \(E\) is $$ L \frac{\mathrm{d} i}{\mathrm{~d} t}+R i=E $$ (a) Solve this equation subject to the condition \(i(0)=0 .\) (b) State the final value of the current. (c) Find the time taken for the current to reach \(95 \%\) of its final value.
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