6:30PM Wed 9 Oct
\( 98 \% \)
H1
1 of 2
8. Let \( a \) be a nonzero real number.
(a) Verify that if \( c \) is an arbitrary constant then
\[
y=(x-c)^{a}
\]
is a solution of
\[
\text { on }(c, \infty) . \quad y^{\prime}=a y^{(a-1) / a}
\]
9. Verify that
\[
y=\left\{\begin{array}{cl}
e^{x}-1, & x \geq 0 \\
1-e^{-x}, & x<0
\end{array}\right.
\]
is a solution of
\[
y^{\prime}=|y|+1
\]
on \( (-\infty, \infty) \). HiNT: Use the definition of derivative at \( x=0 \).
10. (a) Verify that if \( c \) is any real number then
\[
y=c^{2}+c x+2 c+1
\]
satisfies
\[
y^{\prime}=\frac{-(x+2)+\sqrt{x^{2}+4 x+4 y}}{2}
\]
on some open interval. Identify the open interval.
(b) Verify that
\[
y_{1}=\frac{-x(x+4)}{4}
\]
also satisfies (B) on some open interval, and identify the open interval. (Note that \( y_{1} \) can't be obtained by selecting a value of \( c \) in (A).)
In Exercises 1-6 find all solutions.
(A)
1. \( y^{\prime}=\frac{3 x^{2}+2 x+1}{y-2} \)
2. \( (\sin x)(\sin y)+(\cos y) y^{\prime}=0 \)
3. \( x y^{\prime}+y^{2}+y=0 \)
4. \( y^{\prime} \ln |y|+x^{2} y=0 \)
5. \( \left(3 y^{3}+3 y \cos y+1\right) y^{\prime}+\frac{(2 x+1) y}{1+x^{2}}=0 \)
In Exercises 11 and 12 solve the initial value problem.
11. \( y^{\prime}=\frac{x^{2}+3 x+2}{y-2}, \quad y(1)=4 \)
12. \( y^{\prime}+x\left(y^{2}+y\right)=0, \quad y(2)=1 \)
In Exercises 1-17 determine which equations are exact and solve them.
1. \( 6 x^{2} y^{2} d x+4 x^{3} y d y=0 \)
2. \( \left(3 y \cos x+4 x e^{x}+2 x^{2} e^{x}\right) d x+(3 \sin x+3) d y=0 \)
11. \( \left(\frac{1}{x}+2 x\right) d x+\left(\frac{1}{y}+2 y\right) d y=0 \)
12. \( \left(y \sin x y+x y^{2} \cos x y\right) d x+\left(x \sin x y+x y^{2} \cos x y\right) d y=0 \)
13. \( \frac{x d x}{\left(x^{2}+y^{2}\right)^{3 / 2}}+\frac{y d y}{\left(x^{2}+y^{2}\right)^{3 / 2}}=0 \)
In Exercises 3-16 find an integrating factor; that is a function of only one variable, and solve the given equation.
8. \( \left(27 x y^{2}+8 y^{3}\right) d x+\left(18 x^{2} y+12 x y^{2}\right) d y=0 \)
9. \( \left(6 x y^{2}+2 y\right) d x+\left(12 x^{2} y+6 x+3\right) d y=0 \)
10. \( y^{2} d x+\left(x y^{2}+3 x y+\frac{1}{y}\right) d y=0 \)
12. \( \left(x^{2} y+4 x y+2 y\right) d x+\left(x^{2}+x\right) d y=0 \)
16. \( y \sin y d x+x(\sin y-y \cos y) d y=0 \)
