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} (dy)/(dx)=2x^5+5x^4 \\ \end{gathered}`

Find out the value of the function y

`y=\int(dy)/(dx)dx=\int(2x^5+5x^4)dx=(x^6)/(3)+x^5+C`

so

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