Question:
The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.
The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.
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For second-order linear differential equations, check if the equation has a nonzero term on the right-hand side. If it does, the equation is nonhomogeneous.
Updated On: Apr 14, 2025
- second order, nonhomogeneous, and linear
- second order, homogeneous, and linear
- second order, homogeneous, and nonlinear
- first order, nonhomogeneous, and linear
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The Correct Option is A
Solution and Explanation
Step 1: Identifying the order of the differential equation
The equation involves the second derivative y'', so it is a second-order differential equation.
Step 2: Checking if the equation is homogeneous or nonhomogeneous
The right-hand side of the equation is r(x), which is a non-zero function. For the equation to be homogeneous, the right-hand side must be zero. Since r(x) ≠ 0, the equation is nonhomogeneous.
Step 3: Determining if the equation is linear or nonlinear
The equation is linear because y and its derivatives (y' and y'') appear to the first power and are not multiplied together or raised to any powers.
Thus, the correct classification is: second-order, nonhomogeneous, and linear.
The equation involves the second derivative y'', so it is a second-order differential equation.
Step 2: Checking if the equation is homogeneous or nonhomogeneous
The right-hand side of the equation is r(x), which is a non-zero function. For the equation to be homogeneous, the right-hand side must be zero. Since r(x) ≠ 0, the equation is nonhomogeneous.
Step 3: Determining if the equation is linear or nonlinear
The equation is linear because y and its derivatives (y' and y'') appear to the first power and are not multiplied together or raised to any powers.
Thus, the correct classification is: second-order, nonhomogeneous, and linear.
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