Solve the differential equation. xdydx−4y=2x4ex

Question

Solve the differential equation.  xdydx−4y=2x4ex

berggansS · Accepted Answer

Step 1
According to the given information, it is required to solve the differential equation.

x \frac{d y}{d x} - 4 y = 2 x^{4} e^{x}

Step 2
First divide the whole differential equation by x to get the linear differential form.

\frac{d y}{d x} - \frac{4 y}{x} = 2 x^{3} e^{x}

Step 3
Now, the general linear differential equation and its solution is:

\frac{d y}{d x} + P (x) y = Q (x)

solution of differential equation is:

y (x) \times I F = \int Q (x) . I F d x + c

where

I F = e^{\int P (x) d x}

Step 4
Now, solve the given using the above definition.
In the given differential equation is:

P (x) = \frac{- 4}{x}, Q (x) = 2 x^{3} e^{x}

I F = e^{P d x} = e^{\int - \frac{4}{x} d x} = e^{- 4 \log (x)} = e^{\log (x^{- 4})} = x^{- 4}

I F = \frac{1}{x^{4}}

the solution of the given differential equation is:

y (\frac{1}{x^{4}}) = \int 2 x^{3} e^{x} (\frac{1}{x^{4}}) d x + c

\frac{y}{x^{4}} = 2 \int \frac{e^{x}}{x} d x + c

\frac{y}{x^{4}} = 2 E_{i} (x) + c (\int \frac{e^{x}}{x} d x = E_{i} (x) + c)

y (x) = 2 x^{4} E_{i} (x) + c x^{4}

Solve the differential equation. x\frac{dy}{dx}-4y=2x^{4}e^{x}