1st order nonhomogeneous dif'l e'n

[rbwang1225]

f'+x*f=x^(2m+1), m is an integer

I have no idea for finding particular solution.

Any help would be appreciated.

 

[HallsofIvy]

Well, that is a linear equation so it should be fairly easy to find an integrating factor.

That is, you want to find a function, u(x), such that multiplying by it makes the left side an "exact derivative":
$$u(x)\frac{df}{dx}+ xu(x)f(x)= \frac{d(u(x)f}{dx}$$

By the product rule
$$\frac{d(uf)}{dx}= u\frac{df}{dx}+ \frac{du}{dx}f$$

Comparing those, we must have
$$\frac{du}{dx}= xu$$
or
$$\frac{du}{u}= x dx$$

Integrate that to find u(x), then multiply the entire differential equation by u(x) to get an exact equation.

By the way, you really need to use parenthese to clarify. Is that right hand side
(x^2)(m+1) or x^(2m)+ 1 or x^(2m+1)?

(I expect it is the last since that exponent being odd is important in integrating.)

 

[tiny-tim]

hi rbwang1225!

(try using the X² icon just above the Reply box )

f'+x*f=x^2m+1, m is an integer

I have no idea for finding particular solution.

Any help would be appreciated.

is that f' + x*f = x^2m+1 or f' + x*f = x^2m + 1 ?

either way, you have a polynomial on the RHS, so the obvious guess for a particular solution is … ?

 

[pashav46]

rbwang1225
The particular solution is

f*(x)=x^(2m)-2*m*x^(2m-2)+(2^2)*m*(m-1)*x^(2m-4)-...+((-1)^m)*(2^m)*m!

 

[blue_raver22]

I understand the difficulty of solving a first-order nonhomogeneous differential equation with a variable coefficient. However, there are several methods that can be used to find a particular solution to this type of equation.

One approach is to use the method of undetermined coefficients, where we assume a form for the particular solution and then solve for the coefficients by plugging it into the equation. Another method is the variation of parameters, where we assume the particular solution to be a linear combination of the solutions to the homogeneous equation and then solve for the parameters.

In this specific case, where the coefficient is a function of x, we can also use the method of integrating factors. This involves multiplying both sides of the equation by a suitable function, known as the integrating factor, which can help to simplify the equation and make it easier to solve.

I would recommend consulting with a mathematics expert or using a computer program to assist in solving this type of equation. It is also important to carefully check the solution obtained, as there may be multiple solutions or special cases to consider. Overall, it is important to approach this problem with patience and perseverance, as finding a particular solution to a nonhomogeneous differential equation can be a challenging but rewarding task.