2.6E: Integrating Factors (Exercises)

Q2.6.1

1.

1. Verify that $\mu(x,y)=y$ is an integrating factor for $$y dx+\left(2x+\frac{1}{y} \right) dy=0 \tag{A}$$ on any open rectangle that does not intersect the $x$ axis or, equivalently, that $$y^{2} dx +(2xy+1) dy=0 \tag{B}$$ is exact on any such rectangle.
2. Verify that $y\equiv0$ is a solution of (B), but not of (A).
3. Show that $$y(xy+1)=c \tag{C}$$ is an implicit solution of (B), and explain why every differentiable function $y=y(x)$ other than $y\equiv0$ that satisfies (C) is also a solution of (A).

2.

1. Verify that $\mu(x,y)=1/(x-y)^2$ is an integrating factor for $$-y^{2}dx+x^{2}dy=0 \tag{A}$$ on any open rectangle that does not intersect the line $y=x$ or, equivalently, that $$-\frac{y^{2}}{(x-y)^{2}}dx + \frac{x^{2}}{(x-y)^{2}}dy=0 \tag{B}$$ is exact on any such rectangle.
2. Use Theorem 2.2.1 to show that $$\frac{xy}{(x-y)}=c\tag{C}$$ is an implicit solution of (B), and explain why it is also an implicit solution of (A)
3. Verify that $y=x$ is a solution of (A), even though it can’t be obtained from (C).

Q2.6.2

In Exercises 2.6.3-2.6.16 find an integrating factor; that is a function of only one variable, and solve the given equation.

3. $ydx-xdy=0$

4. $3x^{2}ydx +2x^{3}dy=0$

5. $2y^{3}dx+3y^{2}dy=0$

6. $(5xy+2y+5)dx+2xdy=0$

7. $(xy+x+2y+1)\,dx+(x+1)\,dy=0$

8. $(27xy^2+8y^3)\,dx+(18x^2y+12xy^2)\,dy=0$

9. $(6xy^2+2y)\,dx+(12x^2y+6x+3)\,dy=0$

10. $y^2\,dx+\left(xy^2+3xy+{1\over y}\right)\,dy=0$

11. $(12x^3y+24x^2y^2)\,dx+(9x^4+32x^3y+4y)\,dy=0$

12. $(x^2y+4xy+2y)\,dx+(x^2+x)\,dy=0$

13. $-y\,dx+(x^4-x)\,dy=0$

14. $\cos x\cos y\,dx +(\sin x\cos y-\sin x\sin y+y)\,dy=0$

15. $(2xy+y^2)\,dx+(2xy+x^2-2x^2y^2-2xy^3)\,dy=0$

16. $y\sin y\,dx+x(\sin y-y\cos y)\,dy=0$

Q2.6.3

In Exercises 2.6.17-2.6.23 find an integrating factor of the form $\mu (x,y)=P(x)Q(y)$ and solve the given equation.

17. $y(1+5\ln|x|)\,dx+4x\ln|x|\,dy=0$

18. $(\alpha y+ \gamma xy)\,dx+(\beta x+ \delta xy)\,dy=0$

19. $(3x^2y^3-y^2+y)\,dx+(-xy+2x)\,dy=0$

20. $2y\,dx+ 3(x^2+x^2y^3)\,dy=0$

21. $(a\cos xy-y\sin xy)\,dx+(b\cos xy-x\sin xy)\, dy=0$

22. $x^4y^4\,dx+x^5y^3\,dy=0$

23. $y(x\cos x+2\sin x)\,dx+x(y+1)\sin x\,dy=0$

Q2.6.4

In Exercises 2.6.24-2.6.27 find an integrating factor and solve the equation. Plot a direction field and some integral curves for the equation in the indicated rectangular region.

24. $(x^4y^3+y)\,dx+(x^5y^2-x)\,dy=0; \quad \{-1\le x\le1,-1\le y\le1\}$

25. $(3xy+2y^2+y)\,dx+(x^2+2xy+x+2y)\,dy=0; \quad \{-2\le x\le2,-2\le y\le2\}$

26. $(12 xy+6y^3)\,dx+(9x^2+10xy^2)\,dy=0; \quad \{-2\le x\le2,-2\le y\le2\}$

27. $(3x^2y^2+2y)\,dx+ 2x\,dy=0; \quad \{-4\le x\le4,-4\le y\le4\}$

Q2.6.5

28. Suppose $a$, $b$, $c$, and $d$ are constants such that $ad-bc\ne0$, and let $m$ and $n$ be arbitrary real numbers. Show that

$$(ax^my+by^{n+1})\,dx+(cx^{m+1}+dxy^n)\,dy=0 \nonumber$$

has an integrating factor $\mu(x,y)=x^\alpha y^\beta$.

29. Suppose $M$, $N$, $M_x$, and $N_y$ are continuous for all $(x,y)$, and $\mu=\mu(x,y)$ is an integrating factor for $$M(x,y)dx+N(x,y)dy=0.\tag{A}$$

Assume that $\mu_x$ and $\mu_y$ are continuous for all $(x,y)$, and suppose $y=y(x)$ is a differentiable function such that $\mu(x,y(x))=0$ and $\mu_x(x,y(x))\ne0$ for all $x$ in some interval $I$. Show that $y$ is a solution of (A) on $I$.

30. According to Theorem 2.1.2, the general solution of the linear nonhomogeneous equation $$y'+p(x)y=f(x)\tag{A}$$

is $$y=y_{1}x\left( c+\int f(x)/y_{1}(x) dx \right),\tag{B}$$

where $y_1$ is any nontrivial solution of the complementary equation $y'+p(x)y=0$. In this exercise we obtain this conclusion in a different way. You may find it instructive to apply the method suggested here to solve some of the exercises in Section 2.1.

1. Rewrite (A) as $$[p(x)y-f(x)]dx +dy =0,\tag{C}$$ and show that $\mu=\pm e^{\int p(x)\,dx}$ is an integrating factor for (C).
2. Multiply (A) through by $\mu=\pm e^{\int p(x)\,dx}$ and verify that the resulting equation can be rewritten as $$(\mu(x)y)'=\mu(x)f(x). \nonumber$$ Then integrate both sides of this equation and solve for $y$ to show that the general solution of (A) is $$y={1\over\mu(x)}\left(c+\int f(x)\mu(x)\,dx\right). \nonumber$$ Why is this form of the general solution equivalent to (B)?