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# 2.6E: Integrating Factors (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## 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

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

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 $(\mu(x)y)'=\mu(x)f(x).$ Then integrate both sides of this equation and solve for $$y$$ to show that the general solution of (A) is $y={1\over\mu(x)}\left(c+\int f(x)\mu(x)\,dx\right).$ Why is this form of the general solution equivalent to (B)?