Solve the initial value problem: d²y/dt² = 1 - e^(-2t), y(1) = -1, y'(1) = 4.

Steps to Solve

1. Integrate the Differential Equation

To solve the second-order differential equation $$ \frac{d^{2}y}{dt^{2}} = 1 - e^{-2t} $$, we first integrate it with respect to ( t ).

$$ \frac{dy}{dt} = \int (1 - e^{-2t}) , dt $$

The integral can be computed as:

$$ \frac{dy}{dt} = t + \frac{1}{2}e^{-2t} + C_1 $$

where ( C_1 ) is the constant of integration.

2. Second Integration

Next, we integrate again to find ( y(t) ):

$$ y(t) = \int \left(t + \frac{1}{2}e^{-2t} + C_1\right) dt $$

Calculating the integral gives:

$$ y(t) = \frac{t^2}{2} - \frac{1}{4}e^{-2t} + C_1t + C_2 $$

where ( C_2 ) is another constant of integration.

3. Applying Initial Conditions

Now, we apply the initial conditions ( y(1) = -1 ) and ( y'(1) = 4 ).

First, calculate ( y(1) ): $$ -1 = \frac{1^2}{2} - \frac{1}{4}e^{-2} + C_1 \cdot 1 + C_2 $$

This simplifies to: $$ -1 = \frac{1}{2} - \frac{1}{4}e^{-2} + C_1 + C_2 $$

Next, calculate ( y'(1) ): $$ y'(1) = 1 + \frac{1}{2}(-2)e^{-2} + C_1 = 4 $$

This simplifies to: $$ 1 - e^{-2} + C_1 = 4 $$

4. Solving for Constants

From ( 1 - e^{-2} + C_1 = 4 ): $$ C_1 = 4 - 1 + e^{-2} = 3 + e^{-2} $$

Substituting ( C_1 ) into the first condition:

$$ -1 = \frac{1}{2} - \frac{1}{4}e^{-2} + (3 + e^{-2}) + C_2 $$

This simplifies to: $$ C_2 = -1 - \frac{1}{2} + \frac{1}{4}e^{-2} - 3 - e^{-2} $$

5. Final Equation

Thus, we can plug back the values of ( C_1 ) and ( C_2 ) to obtain the complete solution.

The final function ( y(t) ) becomes:

$$ y(t) = \frac{t^2}{2} - \frac{1}{4}e^{-2t} + (3 + e^{-2})t + C_2 $$

6. Final Simplification

You will compute the exact values for ( C_2 ) finally using the obtained expressions and simplify ( y(t) ).