Question

find the particular antiderivative that satisfies the following conditions: $\frac{dy}{dx}=\frac{7x + 3}{sqrt3{x}};y(1)=2$.

Explanation:

Step1: Rewrite the derivative

We have $\frac{dy}{dx}=\frac{7x + 3}{\sqrt[3]{x}}=(7x + 3)x^{-\frac{1}{3}}=7x^{\frac{2}{3}}+3x^{-\frac{1}{3}}$.

Step2: Integrate term - by - term

Using the power rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), we get $\int(7x^{\frac{2}{3}}+3x^{-\frac{1}{3}})dx=7\int x^{\frac{2}{3}}dx+3\int x^{-\frac{1}{3}}dx$.
$7\times\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}+3\times\frac{x^{-\frac{1}{3}+1}}{-\frac{1}{3}+1}=7\times\frac{x^{\frac{5}{3}}}{\frac{5}{3}}+3\times\frac{x^{\frac{2}{3}}}{\frac{2}{3}}=\frac{21}{5}x^{\frac{5}{3}}+\frac{9}{2}x^{\frac{2}{3}}+C$. So $y=\frac{21}{5}x^{\frac{5}{3}}+\frac{9}{2}x^{\frac{2}{3}}+C$.

Step3: Use the initial condition

Given $y(1) = 2$, substitute $x = 1$ and $y=2$ into $y=\frac{21}{5}x^{\frac{5}{3}}+\frac{9}{2}x^{\frac{2}{3}}+C$.
$2=\frac{21}{5}\times1^{\frac{5}{3}}+\frac{9}{2}\times1^{\frac{2}{3}}+C$.
$2=\frac{21}{5}+\frac{9}{2}+C$.
First, find a common denominator: $\frac{21}{5}+\frac{9}{2}=\frac{21\times2}{5\times2}+\frac{9\times5}{2\times5}=\frac{42}{10}+\frac{45}{10}=\frac{42 + 45}{10}=\frac{87}{10}$.
Then $C=2-\frac{87}{10}=\frac{20 - 87}{10}=-\frac{67}{10}$.

Step4: Write the particular antiderivative

$y=\frac{21}{5}x^{\frac{5}{3}}+\frac{9}{2}x^{\frac{2}{3}}-\frac{67}{10}$.

Answer:

$y=\frac{21}{5}x^{\frac{5}{3}}+\frac{9}{2}x^{\frac{2}{3}}-\frac{67}{10}$