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

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

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)?