Question

Solve . d²Bz(x)/dx²=Bz(x)/ some contant²
The z is in subscript and solve this equation.

Answer 1

Answer on Question #62780 – Math – Differential Equations

Question

Solve

$$\frac{d^2Bz(x)}{dx^2} = Bz(x) \quad \text{some constant}^2$$

The $z$ is in subscript and solve this equation.

Solution

Let $a^2$ be a constant. The solution of equation

$$\frac{d^2B_z(x)}{dx^2} = \frac{B_z(x)}{a^2},$$

can be found in the following form:

$$B_z(x) = e^{\lambda x}.$$

Substituting (2) into the equation (1) obtain

$$\frac{d^2B_z(x)}{dx^2} = \lambda^2 e^{\lambda x} = \frac{1}{a^2} e^{\lambda x}.$$

From (3) we obtain the characteristic equation:

$$\lambda^2 = \frac{1}{a^2}.$$

The solutions of equation (4) are

$$\lambda = \frac{1}{a} \quad \text{and} \quad \lambda = -\frac{1}{a}.$$

Then we obtain two particular solutions, namely

$$B_{z1}(x) = e^{x/a} \quad \text{and} \quad B_{z2}(x) = e^{-x/a}.$$

The general solution of the original equation (1) is a linear combination of particular solutions:

$$B_z(x) = C e^{x/a} + D e^{-x/a},$$

where $C$ and $D$ are real constants.

The general solution of (1) can also be written as

$$B _ {z} (x) = E \sinh (x / a) + F \cosh (x / a),$$

where $E$ and $F$ are real constants;

$\sinh (x / a) = \frac{1}{2}\left(e^{x / a} - e^{-x / a}\right)$ is the hyperbolic sine and $\cosh (x / a) = \frac{1}{2}\left(e^{x / a} + e^{-x / a}\right)$ is the hyperbolic cosine. These functions are particular solutions too.

Answer: $B_{z}(x) = Ce^{x / a} + De^{-x / a}$ or

$$B _ {z} (x) = E \sinh (x / a) + F \cosh (x / a).$$

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