Solve differential equation with variation of parameters

[plucker_08]

solve using method of variation of parameters

y''-y = 2/(1+e^x)

y'' ==> second order

 

[HallsofIvy]

And your question is?

It should be easy to see that e^x and e^-x are two independent solutions to y"- y= 0.

The "variation of parameters" is to look for a solution of the form
y(x)= u(x)e^x+ v(x)e^-x
In fact, there are an infinite number of u(x), v(x) that would work.

Differentiating, y'= u'(x)e^x+ u(x)e^x+ v'(x)e^-x- v(x)e^-x.

ASSUME that u'(x)e^x+ v'(x)e^-x= 0.
(Since there are an infinite number of u(x), v(x) that would work above, this is just "narrowing the search".)

With that assumption y'= u(x)e^x- v(x)e^-x.

Differentiating again, y"= u'(x)e^x+ u(x)e^x- v'(x)e^-x+ ve^-x.

Plug that into the original equation and, because e^x and e^-x satisfy the original homogeneous equation, the "u(x)" and "v(x)" terms cancel leaving just u'e^x- v'e^-x= 2/(1+e^x). Treat that, along with
u'e^x+ v'e^-x= 0 (above) as two linear equations for u', v'.
Integrate those solutions to find u and v and plug into y(x)= u(x)e^x+ v'(x)e^-x.

 

[quicknote]

differential equation
y ==> dependent variable
x ==> independent variable

The method of variation of parameters is a technique used to solve second order differential equations, such as the one given in this problem. It involves finding a particular solution by varying the parameters of the general solution. In this case, the general solution of the differential equation is y = c1e^x + c2e^-x, where c1 and c2 are arbitrary constants.

To use the method of variation of parameters, we first need to find the complementary solution, which is the general solution without the term involving the dependent variable. In this case, the complementary solution is y_c = c1e^x + c2e^-x.

Next, we need to find the particular solution, which involves finding two functions u1(x) and u2(x) that will be multiplied by c1 and c2 respectively, to create the particular solution. These functions can be found by solving a system of equations involving the derivatives of u1 and u2.

In this case, we have u1' = 0 and u2' = 2e^x/(1+e^x). Solving these equations, we get u1 = 1 and u2 = 2x.

Therefore, the particular solution is y_p = c1 + 2xc2.

The general solution of the differential equation is then y = y_c + y_p = c1e^x + c2e^-x + c1 + 2xc2.

To find the specific solution, we need to use initial conditions or boundary conditions. For example, if y(0) = 1 and y'(0) = 0, we can solve for c1 and c2 and get the specific solution y = e^x + 2xe^-x.

In summary, the method of variation of parameters is a powerful tool for solving second order differential equations with non-constant coefficients. It involves finding a complementary solution and a particular solution, and then combining them to get the general solution. It is a useful technique for scientists and engineers in various fields, as it allows for the solution of complex equations that arise in many real-world applications.