Use the scale factor of corresponding sides to determine the value of y.

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nalex0528

25.02.2020

Mathematics

The answer is -3.215======this is a separable differential equation problem. it gives us an initial value to work worth, meaning we have to find a specific solution to the differential equation. then we use that solution curve to find the value of y at x = π $\dfrac{dy}{dx} = x \cos \left( x^2 \right)$ move the differential dx to the right-hand side since all the x-terms are there. $dy = x \cos \left( x^2 \right) dx$ integrate both sides of the equation. $\displaystyle \int dy = \int x \cos \left( x^2 \right) dx$ using substitution for the right-hand-side integral, we let u = x². thus simple differentiation gives us $du = 2x\, dx \implies dx = \dfrac{du}{2x}$ so substituting it into the integral, we get $\begin{aligned} \int dy & = \int x \cos \left( u \right) \frac{du}{2x} \\ y & = \int \cos \left( u \right) \frac{du}{2} & & \text{\footnotesize $x$ cancel out} \\ y & = \frac{1}{2}\int \cos \left( u \right)du & & \\ y & = \frac{1}{2} \sin(u) + c \\ y & = \frac{1}{2} \sin(x^2) + c & & \text{\footnotesize back-substituting} \end{aligned}$ solve for c using initial condition: "y = -3 when x = 0" $\begin{aligned} -3 & = \frac{1}{2} \sin(0) + c \\ -3 & = \frac{1}{2} (0) + c \\ c & = -3 \end{aligned}$ therefore, the specific solution curve is $y = \frac{1}{2} \sin(x^2) -3$ . when evaluated at x = π, we get a y-value of $\begin{aligned} y(\pi) & = \tfrac{1}{2} \sin(\pi^2 ) -3 \\ & \approx -3.215 \end{aligned}$ the answer is -3.215

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