The text discusses the method of variation of parameters for solving non-homogeneous differential equations. Here's a detailed breakdown:### General Method for Non-Homogeneous Differential EquationsFor a non-homogeneous problem \( y'' + p(x)y' + q(x)y = f(x) \), assume \( y_1, y_2 \) is a fundamental set of solutions for the homogeneous problem \( y'' + p(x)y' + q(x)y = 0 \). The formula for the particular solution using the method of variation of parameters is:\[y_p = y_1 u_1 + y_2 u_2\]where:\[u_1 = -\frac{y_2(x)f(x)}{W(x)} \quad \text{and} \quad u_2 = \frac{y_1(x)f(x)}{W(x)}\]The Wronskian, \( W(x) \), is given by the determinant:\[W(x) = \begin{vmatrix} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \end{vmatrix}\]Therefore, we have:\[u_1 = \int \frac{-y_2(x)f(x)}{W(x)} \, dx \quad \text{and} \quad u_2 = \int \frac{y_1(x)f(x)}{W(x)} \, dx\]**Note**: When evaluating these indefinite integrals, the arbitrary constant of integration is set to zero.The single integral formula is:\[y_p(x) = y_1(x) \int \frac{-y_2(x)f(x)}{W(x)} \, dx + y_2(x) \int \frac{y_1(x)f(x)}{W(x)} \, dx\]### Example ProblemConsider the non-homogeneous problem \( y'' - 9y = 7e^{4x} \).1. **General Solution of the Homogeneous Problem** The complementary solution, \( y_c = ay_1 + by_2 \), is given by linearly independent solutions \( y_1, y_2 \). Here \( a \) and \( b \) are arbitrary constants. - Find a fundamental set

Answered: Image /qna-images/answer/4f0315b3-9327-45b7-a93d-6e54d1f4d33f.jpg

In general for a non-homogeneous problem y" + p(z)y +q(z)y = f(z) assume that y1, 2 is a fundamental set of solutions for the homogeneous problem y" + p(z)y' + g(z)y = 0. Then the formula for the particular solution using the method of variation of parameters is -y2(z)f(z) W(z) Y1 (z)f(z) W(z) where u = and u = where W(z) is the Wronskian given by the determinant |1 (z) 2(z)| W(z) = yn (포)f(z) -2(z)f(x) W(z) So we have u - dz and uz = dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. W(z) In other words we have the single integral formula _2(z)f(z) p(z) = y1(z) | - W(z) dz + y2(z) [ 4(2)f(2) dz W(z) As a specific example we consider the non-homogeneous problem y" – 9y = 7ez (1) The general solution of the homogeneous problem (called the complementary solution, yc = ayı + byz ) is given in terms of a pair of linearly independent solutions, Y1. Y2. Here a and b are arbitrary constants. Find a fundamental set for y" - 9y = 0 and enter your results as a comma separated list e^(3x),e^(-3x) BEWARE Notice that the above set does not require you to decide which function is to be called y, or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and y need to stick to that order. In order to more easily allow WeBWorK to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for yh = e^(3x) a if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2 = e^(-3x) With this appropriate order we are now ready to apply the method of variation of parameters. (2) For our particular problem we have W(z) = -ya(포)f(z) dz = dz = e^(4x) = In W(z) * y1 (z)f(z) dz = u2 = dz = W(z) And combining these results we arrive at Yp (3) Finally, the general solution is y = yc + Yp where ye = ayı+ byz where a and b are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0) = 1 and y'(0) = -2. y = -e^(3x)-2e^(-3x)+e^(4x)