2.1E: Linear First Order Equations (Exercises)

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

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Q2.1.1

In Exercises 2.1.1-2.1.5 find the general solution.

1. $y'+ay=0$ ( $a$=constant)

2. $y'+3x^2y=0$

3. $xy'+(\ln x)y=0$

4. $xy'+3y=0$

5. $x^2y'+y=0$

Q2.1.2

In Exercises 2.1.6-2.1.11 solve the initial value problem.

6. $ {y'+\left({1+x\over x}\right)y=0,\quad y(1)=1}$

37 $ {xy'+\left(1+{1\over\ln x}\right)y=0,\quad y(e)=1}$

8. $ {xy'+(1+ x\cot x)y=0,\quad y\left({\pi\over 2} \right)=2}$

9. $ {y'-\left({2x\over 1+x^2}\right)y=0,\quad y(0)=2}$

10. $ y'+\frac{k}{x}y=0,\quad y(1)=3\quad(k=\text{constant})$

11. $ y'+(\tan kx)y=0,\quad y(0)=2\quad (k=\text{constant})$

Q2.1.3

In Exercises 2.1.12-2.1.15 find the general solution. Also, plot a direction field and some integral curves on the rectangular region $\{−2 ≤ x ≤ 2, −2 ≤ y ≤ 2\}$.

12. $y'+3y=1$

13. $ {y'+\left({1\over x}- 1\right)y=-{2\over x}}$

14. $y'+2xy=xe^{-x^2}$

15. $ {y'+{2x\over1+x^2}y={e^{-x}\over1+x^2}}$

Q2.1.4

In Exercises 2.1.16-2.1.24 find the general solution.

16. $ {y'+{1\over x}y={7\over x^2}+3}$

17. $ {y'+{4\over x-1}y = {1\over (x-1)^5}+{\sin x\over (x-1)^4}}$

18. $xy'+(1+2x^2)y=x^3e^{-x^2}$

19. $ {xy'+2y={2\over x^2}+1}$

20. $y'+(\tan x)y=\cos x$

21. $ {(1+x)y'+2y={\sin x \over 1 + x}}$

22. $(x-2)(x-1)y'-(4x-3)y=(x-2)^3$

23. $y'+(2\sin x\cos x) y=e^{-\sin^2x}$

24. $x^2y'+3xy=e^x$

Q2.1.5

In Exercises 2.1.25-2.1.29 solve the initial value problem and sketch the graph of the solution.

25. $y'+7y=e^{3x},\quad y(0)=0$

26. $ {(1+x^2)y'+4xy={2\over 1+x^2},\quad y(0)=1}$

27. $ {xy'+3y={2\over x(1+x^2)},\quad y(-1)=0}$

28. $ {y'+ (\cot x)y=\cos x,\quad y\left({\pi\over 2}\right)=1}$

29. $ {y'+{1\over x}y={2\over x^2}+1,\quad y(-1)=0}$

Q2.1.6

In Exercises 2.1.30-2.1.37, solve the initial value problem.

30. $ {(x-1)y'+3y={1\over (x-1)^3} + {\sin x\over (x-1)^2},\quad y(0)=1}$

31. $xy'+2y=8x^2,\quad y(1)=3$

32. $xy'-2y=-x^2,\quad y(1)=1$

33. $y'+2xy=x,\quad y(0)=3$

34. $ {(x-1)y'+3y={1+(x-1)\sec^2x\over (x-1)^3},\quad y(0)=-1}$

35. $ {(x+2)y'+4y={1+2x^2\over x(x+2)^3},\quad y(-1)=2}$

36. $(x^2-1)y'-2xy=x(x^2-1),\quad y(0)=4$

37. $(x^2-5)y'-2xy=-2x(x^2-5),\quad y(2)=7$

Q2.1.7

In Excercises 2.1.28-2.1.42 solve the initial value problem and leave the answer in a form involving a definite integral. (You can solve these problems numerically by methods discussed in Chapter 3.)

38. $y'+2xy=x^2,\quad y(0)=3$

39. $ {y'+{1\over x}y={\sin x\over x^2},\quad y(1)=2}$

40. $ {y'+y={e^{-x}\tan x\over x},\quad y(1)=0}$

41. $ {y'+{2x\over 1+x^2}y={e^x\over (1+x^2)^2}, \quad y(0)=1}$

42. $xy'+(x+1)y=e^{x^2},\quad y(1)=2$

43. Experiments indicate that glucose is absorbed by the body at a rate proportional to the amount of glucose present in the bloodstream. Let $\lambda$ denote the (positive) constant of proportionality. Now suppose glucose is injected into a patient’s bloodstream at a constant rate of $r$ units per unit of time. Let $G=G(t)$ be the number of units in the patient’s bloodstream at time $t>0$. Then \[G'=-\lambda G+r,\] where the first term on the right is due to the absorption of the glucose by the patient’s body and the second term is due to the injection. Determine $G$ for $t>0$, given that $G(0)=G_0$. Also, find $\lim_{t\to\infty}G(t)$.

44.

(a) Plot a direction field and some integral curves for \[xy'-2y=-1 \tag{A} \] on the rectangular region $\{-1\le x\le 1, -.5\le y\le 1.5\}$. What do all the integral curves have in common?

(b) Show that the general solution of (A) on $(-\infty,0)$ and $(0,\infty)$ is

\[y={1\over2}+cx^2.\]

(c) Show that $y$ is a solution of (A) on $(-\infty,\infty)$ if and only if \[y=\left\{\begin{array}{ll} {{1\over2}+c_1x^2}, &x \ge 0,\\[4pt] {{1\over2}+c_2x^2}, &x < 0,\end{array}\right.\] where $c_1$ and $c_2$ are arbitrary constants.

(d) Conclude from c that all solutions of (A) on $(-\infty,\infty)$ are solutions of the initial value problem \[xy'-2y=-1,\quad y(0)={1\over2}.\]

(e) Use (b) to show that if $x_0\ne0$ and $y_0$ is arbitrary, then the initial value problem \[xy'-2y=-1,\quad y(x_0)=y_0\] has infinitely many solutions on ( $-\infty,\infty$). Explain why this doesn't contradict Theorem 2.1.1.

45. Suppose $f$ is continuous on an open interval $(a,b)$ and $\alpha$ is a constant.

(a) Derive a formula for the solution of the initial value problem

\[y'+\alpha y=f(x),\quad y(x_0)=y_0, \tag{A} \]

where $x_0$ is in $(a,b)$ and $y_0$ is an arbitrary real number.

(b) Suppose $(a,b)=(a,\infty)$, $\alpha > 0$ and $\displaystyle{\lim_{x\to\infty} f(x)=L}$. Show that if $y$ is the solution of (A), then $\displaystyle{\lim_{x\to \infty} y(x)=L/\alpha}$.

46. Assume that all functions in this exercise are defined on a common interval $(a,b)$.

(a) Prove: If $y_1$ and $y_2$ are solutions of

\[y'+p(x)y=f_1(x)\]

and

\[y'+p(x)y=f_2(x)\]

respectively, and $c_1$ and $c_2$ are constants, then $y=c_1y_1+c_2y_2$ is a solution of

\[y'+p(x)y=c_1f_1(x)+c_2f_2(x).\]

(This is the principle of superposition.)

(b) Use (a) to show that if $y_1$ and $y_2$ are solutions of the nonhomogeneous equation

\[y'+p(x)y=f(x), \quad{\rm (A)}\]

then $y_1-y_2$ is a solution of the homogeneous equation

\[y'+p(x)y=0. \quad{\rm (B)}\]

(c) Use (a) to show that if $y_1$ is a solution of (A) and $y_2$ is a solution of (B), then $y_1+y_2$ is a solution of (A).

47. Some nonlinear equations can be transformed into linear equations by changing the dependent variable. Show that if

\[g'(y)y'+p(x)g(y)=f(x)\]

where $y$ is a function of $x$ and $g$ is a function of $y$, then the new dependent variable $z=g(y)$ satisfies the linear equation

\[z'+p(x)z=f(x).\]

48. Solve by the method discussed in Exercise 47.

(a) $(\sec^2y)y'- 3\tan y=-1$

(b) $ {e^{y^2}\left(2yy'+ {2\over x}\right) ={1\over x^2}}$

(d) $ {{y'\over (1+y)^2} - {1\over x(1+y)}=-{3\over x^2}}$

49. We’ve shown that if $p$ and $f$ are continuous on $(a,b)$ then every solution of

\[y'+p(x)y=f(x) \tag{A}\]

on $(a,b)$ can be written as $y=uy_1$, where $y_1$ is a nontrivial solution of the complementary equation for (A) and $u'=f/y_1$. Now suppose $f$, $f'$, …, $f^{(m)}$ and $p$, $p'$, …, $p^{(m-1)}$ are continuous on $(a,b)$, where $m$ is a positive integer, and define

\[\begin{aligned} f_0&=f,\\ f_j&=f_{j-1}'+pf_{j-1},\quad 1\le j\le m.\end{aligned}\]

Show that

\[u^{(j+1)}={f_j\over y_1},\quad 0\le j\le m.\]