Concept explainers
The equation y" + y' – 2y = x2 is called a differential equation because it involves an unknown function y and its derivatives y' and y". Find constants A, B. and C such that the function y = Ax2 + Bx + C satisfies this equation. (Differential equations will he studied in detail in Chapter 9.)
To find: The function y.
Answer to Problem 62E
The function is
Explanation of Solution
Given:
The differential equation is
Derivative rules:
(1) Constant multiple rule:
(2) Power rule:
(3) Sum rule:
Calculation:
Obtain the first derivative of
Apply the sum rule (3) and the constant multiple rule (1),
Since the derivative of constant function is zero, then
Apply the power rule (2) and simplify the terms,
Therefore, the first derivative of
Obtain the second derivative of
Apply the sum rule (3) and the constant multiple rule (1),
Since derivative of constant function is zero, then
Apply the power rule (2) and simplify the terms,
Therefore, the second derivative of
Obtain the required function.
Substitute the values of
The left hand side coefficient of
The left hand side coefficient of x is equal to right hand side coefficient of x.
Substitute
The left hand side of constant term is equal to the right hand side of constant term.
Substitute
Therefore, the value of A, B, and C are
Substitute the value of A, B, C in
Therefore, the required function is
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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