91Ó°ÊÓ

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=-y+2 ; \quad y(t)=2-e^{-t}\)

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=2 t y ; \quad y(t)=e^{t^{2}}\)

In each exercise, (a) Solve the initial value problem analytically, using an appropriate solution technique. (b) For the given initial value problem, write the Heun's method algorithm, $$ y_{n+1}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n}+h f\left(t_{n}, y_{n}\right)\right)\right] . $$ (c) For the given initial value problem, write the modified Euler's method algorithm, $$ y_{n+1}=y_{n}+h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{h}{2} f\left(t_{n}, y_{n}\right)\right) . $$ (d) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (b). (e) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (c). (f) For comparison, calculate and list the exact solution values, \(y\left(t_{1}\right), y\left(t_{2}\right), y\left(t_{3}\right)\). \(y^{\prime}=-y, \quad y(0)=1\)

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=t y^{2} ; \quad y(t)=\frac{2}{2-t^{2}}\)

In each exercise, (a) Solve the initial value problem analytically, using an appropriate solution technique. (b) For the given initial value problem, write the Heun's method algorithm, $$ y_{n+1}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n}+h f\left(t_{n}, y_{n}\right)\right)\right] . $$ (c) For the given initial value problem, write the modified Euler's method algorithm, $$ y_{n+1}=y_{n}+h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{h}{2} f\left(t_{n}, y_{n}\right)\right) . $$ (d) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (b). (e) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (c). (f) For comparison, calculate and list the exact solution values, \(y\left(t_{1}\right), y\left(t_{2}\right), y\left(t_{3}\right)\). \(y^{\prime}=-t y, \quad y(0)=1\)

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=t^{2}+y ; \quad y(t)=3 e^{t}-\left(t^{2}+2 t+2\right)\)

In each exercise, (a) Solve the initial value problem analytically, using an appropriate solution technique. (b) For the given initial value problem, write the Heun's method algorithm, $$ y_{n+1}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n}+h f\left(t_{n}, y_{n}\right)\right)\right] . $$ (c) For the given initial value problem, write the modified Euler's method algorithm, $$ y_{n+1}=y_{n}+h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{h}{2} f\left(t_{n}, y_{n}\right)\right) . $$ (d) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (b). (e) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (c). (f) For comparison, calculate and list the exact solution values, \(y\left(t_{1}\right), y\left(t_{2}\right), y\left(t_{3}\right)\). \(y^{\prime}=-y+t, \quad y(0)=0\)

Assume, for the given differential equation, that \(y(0)=1\). (a) Use the differential equation itself to determine the values \(y^{\prime}(0), y^{\prime \prime}(0), y^{\prime \prime \prime}(0), y^{(4)}(0)\) and form the Taylor polynomial $$ P_{4}(t)=y(0)+y^{\prime}(0) t+\frac{y^{\prime \prime}(0)}{2 !} t^{2}+\frac{y^{\prime \prime \prime}(0)}{3 !} t^{3}+\frac{y^{(4)}(0)}{4 !} t^{4} $$ (b) Verify that the given function is the solution of the initial value problem consisting of the differential equation and initial condition \(y(0)=1\). (c) Evaluate both the exact solution \(y(t)\) and \(P_{4}(t)\) at \(t=0.1\). What is the error \(E(0.1)=y(0.1)-P_{4}(0.1)\) ? [Note that \(E(0.1)\) is the local truncation error incurred in using a Taylor series method of order 4 to step from \(t_{0}=0\) to \(t_{1}=0.1\) using step size \(h=0.1 .]\) \(y^{\prime}=y^{1 / 2} ; \quad y(t)=\left(1+\frac{t}{2}\right)^{2}\)

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=\sqrt{y} ; \quad y(t)=\left(1+\frac{t}{2}\right)^{2}\)

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=\frac{t}{y} ; \quad y(t)=\sqrt{1+t^{2}}\)