Question

Second order ordinary differential equation \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0 \] has values \(y=2\) and \(\frac{dy}{dx} = 1\) at \(x=0\). The value of \(y\) at \(x=1\) is ............ (rounded off to three decimal places).

Hint:

For second-order linear ODEs with constant coefficients: 1. Solve the auxiliary equation. 2. Form the general solution. 3. Apply initial/boundary conditions to find constants. 4. Substitute the required \(x\) value.

Answer 1

Correct Answer: 7.7

Step 1: Write the characteristic equation.
For the given ODE: \[ \frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0 \] Assume solution of form \(y=e^{mx}\). Substituting: \[ m^2 - m - 2 = 0 \] \[ (m-2)(m+1)=0 \Rightarrow m=2, \, m=-1 \]
Step 2: General solution.
\[ y(x) = C_1 e^{2x} + C_2 e^{-x} \]
Step 3: Apply initial conditions.
At \(x=0\), \(y(0)=2\): \[ y(0) = C_1 e^0 + C_2 e^0 = C_1 + C_2 = 2 \Rightarrow (1) \] Now derivative: \[ \frac{dy}{dx} = 2C_1 e^{2x} - C_2 e^{-x} \] At \(x=0\), \(\frac{dy}{dx}=1\): \[ 2C_1 - C_2 = 1 \Rightarrow (2) \]
Step 4: Solve for constants.
From (1): \(C_2 = 2 - C_1\). Substitute into (2): \[ 2C_1 - (2-C_1) = 1 \Rightarrow 2C_1 - 2 + C_1 = 1 \] \[ 3C_1 = 3 \Rightarrow C_1 = 1 \] \[ C_2 = 2 - 1 = 1 \]
Step 5: Final solution.
\[ y(x) = e^{2x} + e^{-x} \]
Step 6: Evaluate at \(x=1\).
\[ y(1) = e^{2} + e^{-1} = 7.389 + 0.368 = 7.757 \] Oops! Let’s check again carefully. (We must recalc because I typed wrong in Final Answer earlier.) At \(x=1\): \[ y(1) = e^{2} + e^{-1} \] \[ = 7.389 + 0.368 = 7.757 \] So final result is: \[ y(1) = 7.757 \, (\text{rounded to three decimals}) \] Final Answer: \[ \boxed{7.757} \]