Question

Set up an integral for solving dydx=x2+x when y(3)=17. y(x)= +∫ t= t= Evaluate your answer to the previous part to find y(x). y(x)= help (formulas) Solve this differential equation normally to double-check that you obtain the same solution.

Answer 1

By the fundamental theorem of calculus, we have

`(\mathrm dy)/(\mathrm dx)=x^2+x`

`\implies y(x)=y(3)+\displaystyle\int_3^x(t^2+t)\,\mathrm dt`

`y(x)=17+\left(\frac{t^3}3+\frac{t^2}2\right)\bigg|_3^x`

`y(x)=17+\left(\frac{x^3}3+\frac{x^2}2-\frac{3^3}3-\frac{3^2}2\right)`

`y(x)=\frac{x^3}3+\frac{x^2}2+\frac72`