Question

solve the differential equation.

\\( \frac { d y } { d x } = 10 x ^ { 4 } e ^ { - y } \\)

the solution is \\( \square \\). (type an equation.)

Explanation:

Step1: Separate variables

We have \(\frac{dy}{dx}=10x^{4}e^{-y}\). Separate the variables: \(e^{y}dy = 10x^{4}dx\).

Step2: Integrate both sides

Integrate \(\int e^{y}dy=\int 10x^{4}dx\).
For the left - hand side, \(\int e^{y}dy=e^{y}+C_1\).
For the right - hand side, using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), we have \(\int 10x^{4}dx=10\times\frac{x^{5}}{5}+C_2 = 2x^{5}+C_2\).
So, \(e^{y}=2x^{5}+C\) (where \(C = C_2 - C_1\)).

Step3: Solve for \(y\)

Take the natural logarithm of both sides: \(y=\ln(2x^{5}+C)\)

Answer:

\(y=\ln(2x^{5}+C)\)