## Related questions with answers

Question

Find the general solution to the exact differential equation. dy/dx = sin x - eˉ˟ + 8x³

Solutions

VerifiedSolution A

Solution B

Answered 2 years ago

Step 1

1 of 3$\dfrac{dy}{dx}=\sin x-e^{-x}+8x^3$

Rewrite the equation in the differential form as shown below:

$dy=\sin x-e^{-x}+8x^3\hspace{1mm}dx$

Integrate both sides

$\int dy=\int \sin x-e^{-x}+8x^3\hspace{1mm}dx$

Step 1

1 of 2The general solution is composed of the antiderivative of the RHS of the exact differential equation. This involves using the power law in reverse. Look for expressions that look like the result of the chain rule.

$\begin{align*} \frac{dy}{dx} &=\sin x - e^{-x} + 8x^3\\ dy&=(\sin x-e^{-x}+8x^3)\;dx\\ \int\;dy&=\int(\sin x-e^{-x}+8x^3)\;dx\\ y&=-\cos x-(-e^{-x})+2x^4+C\\ y &= -\cos x + e^{-x} + 2x^4 \end{align*}$

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