Question

Given dy/dx = 2x^5 + 5x^4 and initial condition y(1) = 2, determine the particular solution.

Answer 1

we know that

$\begin{gathered} \frac{dy}{dx}=2x^5+5x^4 \\ \end{gathered}$

Find out the value of the function y

$y=\int\frac{dy}{dx}dx=\int(2x^5+5x^4)dx=\frac{x^6}{3}+x^5+C$

so

$y=\frac{x^6}{3}+x^5+C$

Remember that

y(1)=2

that means

For x=1, the value of y=2

substitute

$\begin{gathered} 2=\frac{(1)^6}{3}+(1)^5+C \\ 2=\frac{1}{3}+1+C \\ C=\frac{2}{3} \\ \\ therefore \\ The\text{ solution is} \\ y=\frac{x^{6}}{3}+x^5+\frac{2}{3} \end{gathered}$