Solve the initial value problems in Exercises 55–58.


57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0

Verified step by step guidance

1. Identify the given differential equation and initial conditions: \( \frac{d^{2}y}{dx^{2}} = 2e^{-x} \), with \( y(0) = 1 \) and \( y'(0) = 0 \).
2. Integrate the second derivative \( \frac{d^{2}y}{dx^{2}} \) once with respect to \( x \) to find the first derivative \( y'(x) \). This gives \( y'(x) = \int 2e^{-x} \, dx + C_1 \), where \( C_1 \) is a constant of integration.
3. Perform the integration \( \int 2e^{-x} \, dx \) by recalling that \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). Substitute the result back to express \( y'(x) \) including the constant \( C_1 \).
4. Integrate \( y'(x) \) with respect to \( x \) to find \( y(x) \). This will introduce another constant of integration \( C_2 \). So, \( y(x) = \int y'(x) \, dx = \int (\text{expression from previous step}) \, dx + C_2 \).
5. Use the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \) to set up equations and solve for the constants \( C_1 \) and \( C_2 \). Substitute these constants back into the expression for \( y(x) \) to get the particular solution.

Key Concepts

Second-Order Differential Equations

A second-order differential equation involves the second derivative of an unknown function. Solving it requires finding a function y(x) whose second derivative matches the given expression. The general solution often includes two arbitrary constants, reflecting the equation's order.

Initial Conditions and Initial Value Problems

Initial conditions specify the values of a function and its derivatives at a particular point, allowing determination of the arbitrary constants in the general solution. An initial value problem combines a differential equation with these conditions to find a unique solution.

Integration of Exponential Functions

Solving the differential equation involves integrating the right-hand side function, here an exponential function e^(−x). Understanding how to integrate exponentials and apply constants of integration is essential to find the general solution before applying initial conditions.