91Ó°ÊÓ

\(t y^{\prime}+y=t^{2}\)

\(4 \sinh 4 y d y=6 \cosh 3 x d x\)

Consider the IVP \(d y / d x=-y, y(0)=1\). (a) Find the exact solution to the problem. What is \(y(1)\) ? (b) Approximate \(y(1)\) with Euler's method using \(h=0.1, h=0.05\), and \(h=0.025\). (c) Do the values in (b) approach the exact value of \(y(1)\) as \(h\) approaches zero?

(Principle of Superposition for Nonhomogeneous Equations) Show that if \(\phi_{1}=\phi_{1}(t)\) is a solution of \(d y / d t+p(t) y=f_{1}(t)\) and \(\phi_{2}=\phi_{2}(t)\) is a solution of \(d y / d t+p(t) y=\) \(f_{2}(t)\) then \(\phi=\phi_{1}+\phi_{2}\) is a solution of \(d y / d t+p(t) y=f_{1}(t)+f_{2}(t)\).

\(t d y / d t+y=2 t e^{t}, y(1)=-1, t>0\)

\(2 t y^{2} d t+2 t^{2} y d y=0, y(1)=1\)

\(d y / d t+2 y=t^{2} \sqrt{y}, y(0)=1\)

\(\left(e^{y}-2 t y\right) d t+\left(t e^{y}-t^{2}\right) d y=0, y(0)=0\)

\(d y / d x=y / \ln y, y(0)=e\)

\(\left(2 t y+y^{2}\right) d t-t^{2} d y=0\)