6. (a) Solve $$ \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+10 y=0 $$, when $$ \mathrm{y}=4 $$ and $$ \frac{d y}{d x}=1 $$ at $$ \mathrm{x}=0 $$.

Question

UpStudy AI · Accepted Answer

To solve the differential equation:

$$
\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 10 y = 0
$$

with the initial conditions $$ y(0) = 4 $$ and $$ \frac{dy}{dx}(0) = 1 $$, follow these steps:

### 1. Find the General Solution

First, find the characteristic equation associated with the differential equation:

$$
r^2 - 2r + 10 = 0
$$

Solving for $$ r $$:

$$
r = \frac{2 \pm \sqrt{4 - 40}}{2} = \frac{2 \pm \sqrt{-36}}{2} = 1 \pm 3i
$$

Since the roots are complex, the general solution is:

$$
y(x) = e^{x} \left( C_1 \cos(3x) + C_2 \sin(3x) \right)
$$

### 2. Apply Initial Conditions

**First Initial Condition ($$ y(0) = 4 $$):**

$$
y(0) = e^{0} \left( C_1 \cos(0) + C_2 \sin(0) \right) = C_1 = 4
$$

**Second Initial Condition ($$ \frac{dy}{dx}(0) = 1 $$):**

First, compute the derivative of $$ y(x) $$:

$$
\frac{dy}{dx} = e^{x} \left( C_1 \cos(3x) + C_2 \sin(3x) \right) + e^{x} \left( -3C_1 \sin(3x) + 3C_2 \cos(3x) \right)
$$

At $$ x = 0 $$:

$$
\frac{dy}{dx}(0) = C_1 + 3C_2 = 1
$$

Substitute $$ C_1 = 4 $$:

$$
4 + 3C_2 = 1 \implies C_2 = -1
$$

### 3. Write the Particular Solution

Substituting $$ C_1 = 4 $$ and $$ C_2 = -1 $$ back into the general solution:

$$
y(x) = e^{x} \left( 4 \cos(3x) - \sin(3x) \right)
$$

### Final Answer

After simplifying, a particular solution is eˣ times (4 cos 3x minus sin 3x). Thus,

y(x) = e^x [ 4 cos(3x) − sin(3x) ]
The particular solution is $$ y(x) = e^{x} \left( 4 \cos(3x) - \sin(3x) \right) $$.

6. (a) Solve \( \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+10 y=0 \), when \( \mathrm{y}=4 \) and \( \frac{d y}{d x}=1 \) at \( \mathrm{x}=0 \).