91Ó°ÊÓ

Obtain a family of solutions. $$ (x-2 y) d x+(2 x+y) d y=0 $$

Obtain the particular solution satisfying the initial condition indicated. In each exercise interpret your answer in the light of the existence theorem of Section 1.6 and draw a graph of the solution. $$x y y^{\prime}=1+y^{2} ; \text { when } x=2, y=3$$

Determine in each exercise whether or not the function is homogeneous. If it is homogeneous, state the degree of the function. $$ 2 y+\sqrt{x^{2}+y^{2}} $$

Obtain a family of solutions. $$ 2\left(2 x^{2}+y^{2}\right) d x-x y d y=0 $$

Test each of the following equations for exactness and solve the equation. The equations that are not exact may be solved by methods discussed in the preceding sections. \(\left(2 x y-3 x^{2}\right) d x+\left(x^{2}+y\right) d y=0 .\)

Find the general solution. $$ (y+1) d x+(4 x-y) d y=0 $$

Obtain a family of solutions. $$ x y d x-\left(x^{2}+3 y^{2}\right) d y=0 $$

Test each of the following equations for exactness and solve the equation. The equations that are not exact may be solved by methods discussed in the preceding sections. \((2 x y+y) d x+\left(x^{2}-x\right) d y=0\)

Find the general solution. $$ u d x+(1-3 u) x d u=3 u^{2} e^{3 u} d u $$

Obtain the particular solution satisfying the initial condition indicated. In each exercise interpret your answer in the light of the existence theorem of Section 1.6 and draw a graph of the solution. $$2 y d x=3 x d y ; \text { when } x=2, y=1$$