2.4E: Transformation of Nonlinear Equations into Separable Equations (Exercises)

Q2.4.1

In Exercises 2.4.1-2.4.4 solve the given Bernoulli equation.

1. \(y'+y=y^2\)

2. \( {7xy'-2y=-{x^2 \over y^6}}\)

3. \(x^2y'+2y=2e^{1/x}y^{1/2}\)

4. \( {(1+x^2)y'+2xy ={1 \over (1+x^2)y}}\)

Q2.4.2

In Exercises 2.4.5 and 2.4.6 find all solutions. Also, plot a direction field and some integral curves on the indicated rectangular region.

5. \(y'-xy=x^3y^3; \quad \{-3\le x\le 3, -2\le y\le 2\}\)

6. \( {y'-{1+x\over 3x}y=y^4}; \quad \{-2\le x\le2,-2\le y \le2\}\)

Q2.4.3

In Exercises 2.4.7-2.4.11 solve the initial value problem.

7. \(y'-2y=xy^3,\quad y(0)=2\sqrt2\)

8. \(y'-xy=xy^{3/2},\quad y(1)=4\)

9. \(xy'+y=x^4y^4,\quad y(1)=1/2\)

10. \(y'-2y=2y^{1/2},\quad y(0)=1\)

11. \( {y'-4y={48x\over y^2},\quad y(0)=1}\)

Q2.4.4

In Exercises 2.4.12 and 2.4.13 solve the initial value problem and graph the solution.

12. \(x^2y'+2xy=y^3,\quad y(1)=1/\sqrt2\)

13. \(y'-y=xy^{1/2},\quad y(0)=4\)

Q2.4.5

14. You may have noticed that the logistic equation \[P'=aP(1-\alpha P) \nonumber \] from Verhulst’s model for population growth can be written in Bernoulli form as \[P'-aP=-a\alpha P^2. \nonumber \] This isn’t particularly interesting, since the logistic equation is separable, and therefore solvable by the method studied in Section 2.2. So let’s consider a more complicated model, where \(a\) is a positive constant and \(\alpha\) is a positive continuous function of \(t\) on \([0,\infty)\). The equation for this model is \[P'-aP=-a\alpha(t) P^2, \nonumber \] a non-separable Bernoulli equation.

1. Assuming that \(P(0)=P_0>0\), find \(P\) for \(t>0\).
2. Verify that your result reduces to the known results for the Malthusian model where \(\alpha=0\), and the Verhulst model where \(\alpha\) is a nonzero constant.
3. Assuming that \[\lim_{t\to\infty}e^{-at}\int_0^t\alpha(\tau)e^{a\tau}\,d\tau=L \nonumber \] exists (finite or infinite), find \(\lim_{t\to\infty}P(t)\).

Q2.4.6

In Exercises 2.4.15-2.4.18 solve the equation explicitly.

15. \(y'= {y+x\over x}\)

16. \(y'= {y^2+2xy \over x^2}\)

17. \(xy^3y'=y^4+x^4\)

18. \(y'= {y\over x}+\sec{y\over x}\)

Q2.4.7

In Exercises 2.4.19-2.4.21 solve the equation explicitly. Also, plot a direction field and some integral curves on the indicated rectangular region.

19. \(x^2y'=xy+x^2+y^2; \quad \{-8\le x\le 8,-8\le y\le 8\}\)

20. \(xyy'=x^2+2y^2; \quad \{-4\le x\le 4,-4\le y\le 4\}\)

21. \(y'= {2y^2+x^2e^{-(y/x)^2}\over 2xy}; \quad \{-8\le x\le 8,-8\le y\le 8\}\)

Q2.4.8

In Exercises 2.4.22-2.4.27 solve the initial value problem.

22. \(y'= {xy+y^2\over x^2}, \quad y(-1)=2\)

23. \(y'= {x^3+y^3\over xy^2}, \quad y(1)=3\)

24. \(xyy'+x^2+y^2=0, \quad y(1)=2\)

25. \(y'= {y^2-3xy-5x^2 \over x^2}, \quad y(1)=-1\)

26. \(x^2y'=2x^2+y^2+4xy, \quad y(1)=1\)

27. \(xyy'=3x^2+4y^2, \quad y(1)=\sqrt{3}\)

Q2.4.9

In Exercises 2.4.28-2.4.34 solve the given homogeneous equation implicitly.

28. \(y'= {x+y \over x-y}\)

29. \((y'x-y)(\ln |y|-\ln |x|)=x\)

30. \(y'= {y^3+2xy^2+x^2y+x^3\over x(y+x)^2}\)

31. \(y'= {x+2y \over 2x+y}\)

32. \(y'= {y \over y-2x}\)

33. \(y'= {xy^2+2y^3\over x^3+x^2y+xy^2}\)

34. \(y'= {x^3+x^2y+3y^3 \over x^3+3xy^2}\)

Q2.4.10

35.

1. Find a solution of the initial value problem \[x^2y'=y^2+xy-4x^2, \quad y(-1)=0 \tag{A} \] on the interval \((-\infty,0)\). Verify that this solution is actually valid on \((-\infty,\infty)\).
2. Use Theorem 2.3.1 to show that (A) has a unique solution on \((-\infty,0)\).
3. Plot a direction field for the differential equation in (A) on a square \[\{-r\le x\le r, -r\le y\le r\}, \nonumber \] where \(r\) is any positive number. Graph the solution you obtained in (a) on this field.
4. Graph other solutions of (A) that are defined on \((-\infty,\infty)\).
5. Graph other solutions of (A) that are defined only on intervals of the form \((-\infty,a)\), where is a finite positive number.

36.

1. Solve the equation \[xyy'=x^2-xy+y^2 \tag{A} \] implicitly.
2. Plot a direction field for (A) on a square \[\{0\le x\le r,0\le y\le r\} \nonumber \] where \(r\) is any positive number.
3. Let \(K\) be a positive integer. (You may have to try several choices for \(K\).) Graph solutions of the initial value problems \[xyy'=x^2-xy+y^2,\quad y(r/2)={kr\over K}, \nonumber \] for \(k=1\), \(2\), …, \(K\). Based on your observations, find conditions on the positive numbers \(x_0\) and \(y_0\) such that the initial value problem \[xyy'=x^2-xy+y^2,\quad y(x_0)=y_0, \tag{B} \] has a unique solution (i) on \((0,\infty)\) or (ii) only on an interval \((a,\infty)\), where \(a>0\)?
4. What can you say about the graph of the solution of (B) as \(x\to\infty\)? (Again, assume that \(x_0>0\) and \(y_0>0\).)

37.

1. Solve the equation \[y'={2y^2-xy+2x^2 \over xy+2x^2} \tag{A} \] implicitly.
2. Plot a direction field for (A) on a square \[\{-r\le x\le r,-r\le y\le r\} \nonumber \] where \(r\) is any positive number. By graphing solutions of (A), determine necessary and sufficient conditions on \((x_0,y_0)\) such that (A) has a solution on (i) \((-\infty,0)\) or (ii) \((0,\infty)\) such that \(y(x_0)=y_0\).

38. Follow the instructions of Exercise 2.4.37 for the equation \[y'={xy+x^2+y^2 \over xy}. \nonumber \]

39. Pick any nonlinear homogeneous equation \(y'=q(y/x)\) you like, and plot direction fields on the square \(\{-r\le x\le r,\ -r\le y\le r\}\), where \(r>0\). What happens to the direction field as you vary \(r\)? Why?

40. Prove: If \(ad-bc\ne 0\), the equation \[y'={ax+by+\alpha \over cx+dy+\beta} \nonumber \] can be transformed into the homogeneous nonlinear equation \[{dY \over dX}={aX+bY \over cX+dY} \nonumber \] by the substitution \(x=X-X_0,\ y=Y-Y_0\), where \(X_0\) and \(Y_0\) are suitably chosen constants.

Q2.4.11

In Exercises 2.4.21-2.4.43 use a method suggested by Exercise 2.4.40 to solve the given equation implicitly.

41. \(y'= {-6x+y-3 \over 2x-y-1}\)

42. \(y'= {2x+y+1 \over x+2y-4}\)

43. \(y'= {-x+3y-14 \over x+y-2}\)

Q2.4.12

In Exercises 2.4.44-2.4.51 find a function \(y_{1}\) such that the substitution \(y = uy_{1}\) transforms the given equation into a separable equation of the form (2.4.6). Then solve the given equation explicitly.

44. \(3xy^2y'=y^3+x\)

45. \(xyy'=3x^6+6y^2\)

46. \(x^3y'=2(y^2+x^2y-x^4)\)

47. \(y'=y^2e^{-x}+4y+2e^x\)

48. \(y'= {y^2+y\tan x+\tan^2 x\over\sin^2x}\)

49. \(x(\ln x)^2y'=-4(\ln x)^2+y\ln x+y^2\)

50. \(2x(y+2\sqrt x)y'=(y+\sqrt x)^2\)

51. \((y+e^{x^2})y'=2x(y^2+ye^{x^2}+e^{2x^{2}}\)

Q2.4.13

52. Solve the initial value problem \[y'+{2\over x}y={3x^2y^2+6xy+2\over x^2(2xy+3)},\quad y(2)=2. \nonumber \]

53. Solve the initial value problem \[y'+{3\over x}y={3x^4y^2+10x^2y+6\over x^3(2x^2y+5)},\quad y(1)=1. \nonumber \]

54. Prove: If \(y\) is a solution of a homogeneous nonlinear equation \(y'=q(y/x)\), so is \(y_1=y(ax)/a\), where \(a\) is any nonzero constant.

55. A generalized Riccati equation is of the form \[y'=P(x)+Q(x)y+R(x)y^2. \tag{A} \] (If \(R\equiv-1\), (A) is a Riccati equation.) Let \(y_1\) be a known solution and \(y\) an arbitrary solution of (A). Let \(z=y-y_1\). Show that \(z\) is a solution of a Bernoulli equation with \(n=2\).

Q2.4.14

In Exercises 2.4.56-2.4.59, given that \(y_{1}\) is a solution of the given equation, use the method suggested by Exercise 2.4.55 to find other solutions.

56. \(y'=1+x - (1+2x)y+xy^2\); \(y_1=1\)

57. \(y'=e^{2x}+(1-2e^x)y+y^2\); \(y_1=e^x\)

58. \(xy'=2-x+(2x-2)y-xy^2\); \(y_1=1\)

59. \(xy'=x^3+(1-2x^2)y+xy^2\); \(y_1=x\)