if `y=(A+Bx)e^(mx)+(m-1)^-1 e^x` then `(d^2y)/(dx^2)-2m(dy)/(dx)+m^2y` is equal to:

Options

A. `e^(x)`
B. `e^(mx)`
C. `e^(-mx)`
D. `e^((1-m)x)`

Answer

To solve the problem, we need to find the expression for \((d^2y)/(dx^2) - 2m(dy)/(dx) + m^2y\) given \(y = (A + Bx)e^{mx} + (m - 1)^{-1} e^x\).

### Step 1: Find the first derivative \(dy/dx\)

Using the product rule for differentiation, we have:

\[ y = (A + Bx)e^{mx} + (m - 1)^{-1} e^x \]

Let \(u = A + Bx\) and \(v = e^{mx}\).

Using the product rule:

\[ \frac{dy}{dx} = \frac{du}{dx} v + u \frac{dv}{dx} \]

Calculating \(\frac{du}{dx}\) and \(\frac{dv}{dx}\):

\[ \frac{du}{dx} = B \]
\[ \frac{dv}{dx} = me^{mx} \]

Thus,

\[ \frac{dy}{dx} = B e^{mx} + (A + Bx) me^{mx} + (m - 1)^{-1} e^x \]

Combining terms:

\[ \frac{dy}{dx} = (B + m(A + Bx)) e^{mx} + (m - 1)^{-1} e^x \]

### Step 2: Find the second derivative \(d^2y/dx^2\)

Now we differentiate \(\frac{dy}{dx}\):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left((B + m(A + Bx)) e^{mx} + (m - 1)^{-1} e^x\right) \]

Using the product rule again:

1. Differentiate \((B + m(A + Bx)) e^{mx}\):
- Let \(u = B + m(A + Bx)\) and \(v = e^{mx}\).
- Then, \(\frac{du}{dx} = mB\) and \(\frac{dv}{dx} = me^{mx}\).

Thus,

\[ \frac{d}{dx}\left((B + m(A + Bx)) e^{mx}\right) = (mB + m(B + m(A + Bx))) e^{mx} \]

2. Differentiate \((m - 1)^{-1} e^x\):

\[ \frac{d}{dx}\left((m - 1)^{-1} e^x\right) = (m - 1)^{-1} e^x \]

Combining these results, we have:

\[ \frac{d^2y}{dx^2} = (mB + m(B + m(A + Bx))) e^{mx} + (m - 1)^{-1} e^x \]

### Step 3: Substitute into the expression

Now we substitute \(\frac{d^2y}{dx^2}\), \(\frac{dy}{dx}\), and \(y\) into the expression:

\[ \frac{d^2y}{dx^2} - 2m\frac{dy}{dx} + m^2y \]

Substituting the values:

1. Substitute \(\frac{d^2y}{dx^2}\):
\[ (mB + m(B + m(A + Bx))) e^{mx} + (m - 1)^{-1} e^x \]

2. Substitute \(-2m\frac{dy}{dx}\):
\[ -2m\left((B + m(A + Bx)) e^{mx} + (m - 1)^{-1} e^x\right) \]

3. Substitute \(m^2y\):
\[ m^2\left((A + Bx)e^{mx} + (m - 1)^{-1} e^x\right) \]

### Step 4: Combine and simplify

After substituting all parts, we combine like terms. The terms involving \(e^{mx}\) will simplify, and the terms involving \(e^x\) will also simplify.

After simplification, we find that:

\[ \frac{d^2y}{dx^2} - 2m\frac{dy}{dx} + m^2y = e^x \]

### Final Answer

Thus, the final result is:

\[ \frac{d^2y}{dx^2} - 2m\frac{dy}{dx} + m^2y = e^x \]