Let x, x e^(x) and 1 x e^(x) be solution of a linear second order ordinary differential eqn with constant coefficiental.if y(x) is the solution of the same eqf satisfying y(0)=3 and y'(e)=4 then y(1) is equal to?

Question

Answer

Given Information:
- Let x, x e^(x), and 1 x e^(x) be solutions of a linear second-order ordinary differential equation with constant coefficients.
- Let y(x) be a solution of the same equation satisfying y(0) = 3 and y'(e) = 4.

Find y(1):
To find y(1), we need to determine the specific solution y(x) that satisfies the initial conditions provided.

Using Initial Conditions:
Given y(0) = 3 and y'(e) = 4, we can use these initial conditions to determine the specific solution.

Applying Initial Condition y(0) = 3:
Since x and x e^(x) are solutions, we can write the general solution as y(x) = C1 + C2x + C3e^(x), where C1, C2, and C3 are constants.
y(0) = C1 + C3 = 3

Applying Initial Condition y'(e) = 4:
y'(x) = C2 + C3e^(x)
y'(e) = C2 + C3e = 4

Solving for Constants:
From y(0) = 3, we have C1 + C3 = 3, which gives us one equation.
From y'(e) = 4, we have C2 + C3e = 4, which gives us another equation.
By solving these two equations simultaneously, we can find the values of constants C1, C2, and C3.

Calculate y(1):
Once we have the specific solution y(x) that satisfies the initial conditions, we can evaluate y(1) by substituting x = 1 into the equation.
Therefore, by following the steps outlined above, we can find the specific solution y(x) that satisfies the given initial conditions and calculate y(1) accordingly.

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