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Question

# Find a general solution of the differential equation.$dy/dx = 10x^4 - 2x^3$

Solution

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We will find the $\textbf{general solution}$ in the following way. [7pt] We have:

\begin{align*} \dfrac{d y}{d x} &=10 x^{4}-2 x^{3} \quad \Leftrightarrow \quad dy = 10 x^{4}-2 x^{3}\ dx \\[7pt] \int dy &= \int \left(10 x^{4}-2 x^{3}\right)\ dx &\left( \textit{Integrate each side of the equation} \right) \\[7pt] \boldsymbol{y} &= \int \left(10 x^{4}-2 x^{3}\right)\ dx \\[7pt] &= \int 10x^{4}\ dx-\int 2x^{3}\ dx &\left( \textit{Sum Rule } \boldsymbol{\int f(x) \pm g(x) d x=\int f(x) d x \pm \int g(x) d x} \right) \\[7pt] &= 10\int x^{4}\ dx-2\int x^{3}\ dx &\left( \textit{Constant Out } \boldsymbol{\int a \cdot f(x) d x=a \cdot \int f(x) d x} \right) \\[7pt] &= 10\cdot \frac{x^{4+1}}{4+1} - 2\cdot \frac{x^{3+1}}{3+1} &\left( \textit{Power Rule } \boldsymbol{\int x^{a}\ dx=\frac{x^{a+1}}{a+1}} \right) \\[7pt] &= 10\cdot \frac{x^{5}}{5} - 2\cdot \frac{x^{4}}{4} &\left( \textit{Simplify} \right) \\[7pt] &= 2 \cdot x^{5} - \frac{x^{4}}{2} \\[7pt] &= \boldsymbol{\color{#4257b2}2x^5-\dfrac{x^4}{2}+C} &\left( \textit{Add the Constant } \boldsymbol{C} \right) \end{align*}

Therefore, the $\textbf{general solution}$ of the given differential equation is $\boxed{\boldsymbol{y=2x^5-\dfrac{x^4}{2}+C}}$

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