Use the differential equation and the initial condition to find y.

Question

Use the differential equation  and the initial condition  to find y. ​

Quizplus · Accepted Answer

The differential equation given can be rearranged to:
dy/dx = -4y + 7x + 2
This is a first-order linear differential equation, which can be solved by using an integrating factor. Multiplying both sides by e^(-4x), we get:
e^(-4x)dy/dx - 4e^(-4x)y = 7xe^(-4x) + 2e^(-4x)
The left-hand side can be simplified using the product rule:
(d/dx)(e^(-4x)y) = 7xe^(-4x) + 2e^(-4x)
Integrating both sides with respect to x and simplifying, we get:
y = (-7/4)x - (1/8)e^(4x) + C
To find C, we use the initial condition y(0) = 1. Plugging in x = 0 and y = 1, we get:
1 = 0 + C
C = 1
Therefore, the solution to the differential equation with the given initial condition is:
y = (-7/4)x - (1/8)e^(4x) + 1
Simplifying and matching with the given answer choices, we get:
y = 11eac44f_ecee_1e53_bedf_8b89c4a7dbb1_TB7497_11

Use the Differential Equation and the Initial Condition