Question

Consider the differential equation \[ \frac{d^{2}y}{dx^{2}}+3\frac{dy}{dx}+2y=0 \] with the conditions \(y(0)=20\) and \(y'(0)=10\). Then, the value of \[ y(1)e^{2}-50e+20= \] is

• A. \(30\)
• B. \(0\)
• C. \(-30\)
• D. \(-10\)

Hint:

For second-order homogeneous differential equations: \[ \boxed{ am^2+bm+c=0 } \] Find the roots first, write the complementary solution, then use the initial conditions to determine the constants.

Answer 1

The Correct Option is D

Concept: A linear homogeneous differential equation with constant coefficients is solved by first forming its auxiliary equation. For \[ \frac{d^{2}y}{dx^{2}}+3\frac{dy}{dx}+2y=0, \] the auxiliary equation is \[ m^2+3m+2=0. \] The roots determine the complementary solution.

Step 1: Find the complementary solution.
Factorizing, \[ m^2+3m+2=(m+1)(m+2)=0. \] Hence, \[ m=-1,\,-2. \] Therefore, \[ y=C_1e^{-x}+C_2e^{-2x}. \]

Step 2: Use the initial conditions.
Given, \[ y(0)=20. \] Hence, \[ C_1+C_2=20. \] Differentiate, \[ y'=-C_1e^{-x}-2C_2e^{-2x}. \] Using \[ y'(0)=10, \] we obtain \[ -C_1-2C_2=10. \] Solving, \[ C_2=-30, \] \[ C_1=50. \] Thus, \[ \boxed{ y=50e^{-x}-30e^{-2x} } \]

Step 3: Evaluate the required expression.
At \(x=1\), \[ y(1)=\frac{50}{e}-\frac{30}{e^2}. \] Therefore, \[ y(1)e^2 = 50e-30. \] Hence, \[ y(1)e^2-50e+20 = (50e-30)-50e+20 = -10. \] Thus, \[ \boxed{ y(1)e^2-50e+20=-10 } \]

Step 4: Choose the correct option.
\[ \boxed{Option (D) is correct \] \[ \boxed{-10.} \]