Question

Solve the differential equation by variation of parameters. 5y′′−10y′+10y=exsecxy(x)=ex(c1​cos(x)+c2​sin(x))+5ex​cosxlog(cos(x)+xsin(x))​

Answer 1

To solve the given differential equation by variation of parameters, follow these steps: find the complementary solution, assume a particular solution, differentiate and substitute into the equation, solve for the functions, and combine the solutions.

Step-by-step explanation:

To solve the differential equation by variation of parameters, we can follow the steps below:

1. Find the complementary solution by solving the homogeneous equation: 5y'' - 10y' + 10y = 0. This equation has characteristic equation r^2 - 2r + 2 = 0, which gives complex roots r = 1 ± i.
2. Assume a particular solution of the form
   `y_p = u(x)*cos(x) + v(x)*sin(x)` where u(x) and v(x) are functions to be determined.
3. Differentiate y_p to find y_p' and y_p''. Substitute them into the original equation and equate the coefficients of each term.
4. Solve the resulting system of equations to find u(x) and v(x).
5. The general solution is the sum of the complementary solution and the particular solution, y(x) = y_c + y_p.

In this case, the general solution is
`y(x) = e^x(c1*cos(x) + c2*sin(x)) + 5e^x*cos(x)*log(cos(x) + x*sin(x)).`