91Ó°ÊÓ

Find \(y_{3}\) using Picarc's method; also find the exact solution. Compute both \(y_{3}\) and \(y\) for the value of \(x\) shown. $$y^{\prime}=x^{2}-y ; y(1)=3 ; x=0.5$$

Find the Laplace transforms of the given function. $$f(x)=x^{4}$$

Use Picard's method to find the indicated approximation to the solution. $$y^{\prime}=y^{2}-\cos x ; y(0)=1 ; y_{2}$$

Exercises \(1-18\) : Solve the given differential equation. $$y d y-x d x+\left(x^{2}-y^{2}\right) d y=0$$

Find the inverse Laplace transform of the given expression. $$\frac{s}{s^{2}+5}$$

Find the inverse Laplace transform of the given expression. $$\frac{4}{s(s+2)}$$

Find the inverse Laplace transform of the given expression. $$\frac{s-3}{s^{2}+16}$$

The isobars (lines connecting points of equal barometric pressure) on a certain weather map are given by the equation \(y=c / x^{2}\). Find the equation of the orthogonal trajectories that indicate the wind direction from high- to low-pressure areas.