Question

Solve the given differential equation by undetermined coefficients.

y'' + 6y = −294x2e6x
y(x) =
Solve the given differential equation by undetermined coefficients y(x)Solve the given differential equation by undetermined coefficients.
y'' + 6y = −294x2e6x
y(x) =

Answer 1

To solve the given differential equation by undetermined coefficients, assume a particular solution in the form of a sum of polynomial and exponential terms. Find the complementary solution for the homogeneous equation and use the method of undetermined coefficients to find the particular solution. Add the complementary solution and the particular solution to obtain the general solution.

Step-by-step explanation:

To solve the given differential equation by undetermined coefficients, we assume that the particular solution can be expressed as a sum of polynomial and exponential terms. Let's start by finding the complementary solution for the homogeneous equation y'' + 6y = 0. The characteristic equation is r² + 6 = 0, which gives us the roots r = ±i√6.

Therefore, the complementary solution is y_c(x) = c_1cos(√6x) + c_2sin(√6x). Now, for the particular solution, we assume it can be written as y_p(x) = A x² e⁽⁶ˣ⁾. Plugging this into the differential equation, we can determine the values for A and solve for y(x).

Differentiate y_p(x) twice: y''_p(x) = 72A x^2 e⁽⁶ˣ⁾+ 72A x e^{6x} + 12A e⁽⁶ˣ⁾.

Substitute y_p(x) and its derivatives into the differential equation, and collect like terms.

Equate the coefficients of like terms on both sides of the equation and solve for A.

Once you have the value of A, substitute it back into y_p(x) to find the particular solution.

Finally, add the complementary solution y_c(x) and the particular solution y_p(x) to obtain the general solution y(x).

Remember to use proper algebraic manipulations and follow the steps carefully to find the correct particular solution and general solution for the given differential equation.

Answer 2

To solve the given differential equation y'' + 6y = −294x²e⁶x by undetermined coefficients, we assume that the particular solution y(x) has the form A(x)e⁶x.

Step-by-step explanation:

To solve the given differential equation y'' + 6y = −294x²e⁶x by undetermined coefficients, we assume that the particular solution y(x) has the form A(x)e⁶x. Here, A(x) is a polynomial in x of degree 2, which we can find by substituting the assumed form into the differential equation and comparing coefficients. The general solution of the homogeneous equation y'' + 6y = 0 is given by y(x) = c1cos(√(6)x) + c2sin(√(6)x), where c1 and c2 are arbitrary constants. Therefore, the complete solution is y(x) = y_h(x) + y_p(x), where y_h(x) is the solution of the homogeneous equation and y_p(x) is the particular solution.