## Transcription for an Educational Website ### Problem 13 Suppose that one solution \( y_1(x) \) of a homogeneous second-order linear differential equation is known (on an interval \( I \) where \( p \) and \( q \) are continuous functions): \[ y'' + p(x)y' + q(x)y = 0 \] The method of **reduction of order** consists of substituting \( y_2(x) = v(x)y_1(x) \) into the differential equation above and attempting to determine the function \( v(x) \) so that \( y_2(x) \) is a second linearly independent solution of the differential equation. It can be shown that this substitution leads to the following equation, which is a separable equation that is readily solved for the derivative \( v'(x) \) of \( v(x) \). Integration of \( v'(x) \) then gives the desired (nonconstant) function \( v(x) \). \[ y_1v'' + (2y_1' + py_1)v' = 0 \] A differential equation and one solution \( y_1 \) is given below. Use the method of reduction of order to find a second linearly independent solution \( y_2 \): \[ x^2y'' - xy' - 8y = 0; (x > 0); y_1(x) = x^4 \] **Choose the answer below for \( y_2 \) that solves the differential equation and is linearly independent from \( y_1 \):** - A. \( e^{2x} \) - B. \( e^{4x} \) - C. \( \frac{1}{x} \) - D. \( e^x \) - E. \( \frac{1}{x^2} \) - F. \( \frac{1}{x^4} \) ### Problem 14 Suppose that one solution \( y_1(x) \) of a homogeneous second-order linear differential equation is known (on an interval \( I \) where \( p \) and \( q \) are continuous functions): \[ y'' + p(x)y' + q(x)y = 0 \] The method of **reduction of order** consists of substituting \( y_2(x) = v(x)y_1(x)