Q5.1.3
24. Suppose \(p\) and \(q\) are continuous on an open interval \((a,b)\) and let \(x_0\) be in \((a,b)\). Use Theorem 5.1.1 to show that the only solution of the initial value problem
\[y''+p(x)y'+q(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \]
on \((a,b)\) is the trivial solution \(y\equiv0\).
25. Suppose \(P_0\), \(P_1\), and \(P_2\) are continuous on \((a,b)\) and let \(x_0\) be in \((a,b)\). Show that if either of the following statements is true then \(P_0(x)=0\) for some \(x\) in \((a,b)\).
- The initial value problem \[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \] has more than one solution on \((a,b)\).
- The initial value problem \[P_0(x)y''+P_1(x)y'+P_2(x)y=0,\quad y(x_0)=0,\quad y'(x_0)=0\nonumber \] has a nontrivial solution on \((a,b)\).
26. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(y_1\) and \(y_2\) are solutions of
\[y''+p(x)y'+q(x)y=0 \tag{A} \]
on \((a,b)\). Let
\[z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber \]
where \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are constants. Show that if \(\{z_1,z_2\}\) is a fundamental set of solutions of (A) on \((a,b)\) then so is \(\{y_1,y_2\}\).
27. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of
\[y''+p(x)y'+q(x)y=0 \tag{A} \]
on \((a,b)\). Let
\[z_1=\alpha y_1+\beta y_2\quad\text{ and} \quad z_2=\gamma y_1+\delta y_2,\nonumber \]
where \(\alpha,\beta,\gamma\), and \(\delta\) are constants. Show that \(\{z_1,z_2\}\) is a fundamental set of solutions of (A) on \((a,b)\) if and only if \(\alpha\gamma-\beta\delta\ne0\).
28. Suppose \(y_1\) is differentiable on an interval \((a,b)\) and \(y_2=ky_1\), where \(k\) is a constant. Show that the Wronskian of \(\{y_1,y_2\}\) is identically zero on \((a,b)\).
29. Let
\[y_1=x^3\quad\mbox{ and }\quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\[4pt] -x^3,&x<0.\end{array}\right.\nonumber \]
- Show that the Wronskian of \(\{y_1,y_2\}\) is defined and identically zero on \((-\infty,\infty)\).
- Suppose \(a<0<b\). Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).
- Use Exercise 5.1.25b to show that these results don’t contradict Theorem 5.1.5, because neither \(y_1\) nor \(y_2\) can be a solution of an equation \[y''+p(x)y'+q(x)y=0\nonumber \] on \((a,b)\) if \(p\) and \(q\) are continuous on \((a,b)\).
30. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a set of solutions of
\[y''+p(x)y'+q(x)y=0\nonumber \]
on \((a,b)\) such that either \(y_1(x_0)=y_2(x_0)=0\) or \(y_1'(x_0)=y_2'(x_0)=0\) for some \(x_0\) in \((a,b)\). Show that \(\{y_1,y_2\}\) is linearly dependent on \((a,b)\).
31. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of
\[y''+p(x)y'+q(x)y=0\nonumber \]
on \((a,b)\). Show that if \(y_1(x_1)=y_1(x_2)=0\), where \(a<x_1<x_2<b\), then \(y_2(x)=0\) for some \(x\) in \((x_1,x_2)\).
32. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and every solution of
\[y''+p(x)y'+q(x)y=0 \tag{A} \]
on \((a,b)\) can be written as a linear combination of the twice differentiable functions \(\{y_1,y_2\}\). Use Theorem 5.1.1 to show that \(y_1\) and \(y_2\) are themselves solutions of (A) on \((a,b)\).
33. Suppose \(p_1\), \(p_2\), \(q_1\), and \(q_2\) are continuous on \((a,b)\) and the equations
\[y''+p_1(x)y'+q_1(x)y=0 \quad \text{and} \quad y''+p_2(x)y'+q_2(x)y=0\nonumber \]
have the same solutions on \((a,b)\). Show that \(p_1=p_2\) and \(q_1=q_2\) on \((a,b)\).
34. (For this exercise you have to know about \(3\times 3\) determinants.) Show that if \(y_1\) and \(y_2\) are twice continuously differentiable on \((a,b)\) and the Wronskian \(W\) of \(\{y_1,y_2\}\) has no zeros in \((a,b)\) then the equation
\[\frac{1}{W} \left| \begin{array}{ccc} y & y_1 & y_2 \\[4pt] y' & y'_1 & y'_2 \\[4pt] y'' & y_1'' & y_2'' \end{array} \right|=0\nonumber \]
can be written as
\[y''+p(x)y'+q(x)y=0, \tag{A} \]
where \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((a,b)\).
35. Use the method suggested by Exercise 5.1.34 to find a linear homogeneous equation for which the given functions form a fundamental set of solutions on some interval.
- \(e^{x}\cos 2x, e^{x}\sin 2x\)
- \(x, e^{2x}\)
- \(x, x\ln x\)
- \(\cos (\ln x), \sin (\ln x)\)
- \(\cosh x, \sinh x\)
- \(x^{2}-1, x^{2}+1\)
36. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(\{y_1,y_2\}\) is a fundamental set of solutions of
\[y''+p(x)y'+q(x)y=0 \tag{A} \]
on \((a,b)\). Show that if \(y\) is a solution of (A) on \((a,b)\), there’s exactly one way to choose \(c_1\) and \(c_2\) so that \(y=c_1y_1+c_2y_2\) on \((a,b)\).
37. Suppose \(p\) and \(q\) are continuous on \((a,b)\) and \(x_0\) is in \((a,b)\). Let \(y_1\) and \(y_2\) be the solutions of
\[y''+p(x)y'+q(x)y=0 \tag{A} \]
such that
\[y_1(x_0)=1, \quad y'_1(x_0)=0\quad \text{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]
(Theorem 5.1.1 implies that each of these initial value problems has a unique solution on \((a,b)\).)
- Show that \(\{y_1,y_2\}\) is linearly independent on \((a,b)\).
- Show that an arbitrary solution \(y\) of (A) on \((a,b)\) can be written as \(y=y(x_0)y_1+y'(x_0)y_2\).
- Express the solution of the initial value problem \[y''+p(x)y'+q(x)y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \] as a linear combination of \(y_1\) and \(y_2\).
38. Find solutions \(y_1\) and \(y_2\) of the equation \(y''=0\) that satisfy the initial conditions
\[y_1(x_0)=1, \quad y'_1(x_0)=0 \quad \text{and} \quad y_2(x_0)=0, \quad y'_2(x_0)=1.\nonumber \]
Then use Exercise 5.1.37 (c) to write the solution of the initial value problem
\[y''=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]
as a linear combination of \(y_1\) and \(y_2\).
39. Let \(x_0\) be an arbitrary real number. Given (Example 5.1.1) that \(e^x\) and \(e^{-x}\) are solutions of \(y''-y=0\), find solutions \(y_1\) and \(y_2\) of \(y''-y=0\) such that
\[y_1(x_0)=1, \quad y'_1(x_0)=0\quad \text{and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]
Then use
Exercise 5.1.37 (c) to write the solution of the initial value problem
\[y''-y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]
as a linear combination of \(y_1\) and \(y_2\).
40. Let \(x_0\) be an arbitrary real number. Given (Example 5.1.2) that \(\cos\omega x\) and \(\sin\omega x\) are solutions of \(y''+\omega^2y=0\), find solutions of \(y''+\omega^2y=0\) such that
\[y_1(x_0)=1, \quad y'_1(x_0)=0\quad\text{ and} \quad y_2(x_0)=0,\; y'_2(x_0)=1.\nonumber \]
Then use
Exercise 5.1.37 (c) to write the solution of the initial value problem
\[y''+\omega^2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \]
as a linear combination of \(y_1\) and \(y_2\). Use the identities
\[\begin{aligned} \cos(A+B)&=\cos A\cos B-\sin A\sin B\\[4pt] \sin(A+B)&=\sin A\cos B+\cos A\sin B\end{aligned}\nonumber \]
to simplify your expressions for \(y_1\), \(y_2\), and \(y\).
41. Recall from Exercise 5.1.4 that \(1/(x-1)\) and \(1/(x+1)\) are solutions of
\[(x^2-1)y''+4xy'+2y=0 \tag{A} \]
on \((-1,1)\). Find solutions of (A) such that
\[y_1(0)=1, \quad y'_1(0)=0\quad \text{and} \quad y_2(0)=0,\; y'_2(0)=1.\nonumber \]
Then use
Exercise 5.1.37 (c) to write the solution of initial value problem
\[(x^2-1)y''+4xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \]
as a linear combination of \(y_1\) and \(y_2\).
42.
- Verify that \(y_1=x^2\) and \(y_2=x^3\) satisfy \[x^2y''-4xy'+6y=0 \tag{A} \] on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
- Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that \[y=\left\{\begin{array}{rr} a_1x^2+a_2x^3,&x\ge 0,\\[4pt] b_1x^2+b_2x^3,&x<0\phantom{,} \end{array}\right.\nonumber \] is a solution of (A) on \((-\infty,\infty)\) if and only if \(a_1=b_1\). From this, justify the statement that \(y\) is a solution of (A) on \((-\infty,\infty)\) if and only if \[y=\left\{\begin{array}{rr} c_1x^2+c_2x^3,&x\ge 0,\\[4pt] c_1x^2+c_3x^3,&x<0, \end{array}\right.\nonumber \] where \(c_1\), \(c_2\), and \(c_3\) are arbitrary constants.
- For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-4xy'+6y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \] have a solution? What are the solutions?
- Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants, the initial value problem \[x^2y''-4xy'+6y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B} \] has infinitely many solutions on \((-\infty,\infty)\). On what interval does (B) have a unique solution?
43.
- Verify that \(y_1=x\) and \(y_2=x^2\) satisfy \[x^2y''-2xy'+2y=0 \tag{A} \] on \((-\infty,\infty)\) and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
- Let \(a_1\), \(a_2\), \(b_1\), and \(b_2\) be constants. Show that \[y=\left\{\begin{array}{rr} a_1x+a_2x^2,&x\ge 0,\\[4pt] b_1x+b_2x^2,&x<0\phantom{,} \end{array}\right.\nonumber \] is a solution of (A) on \((-\infty,\infty)\) if and only if \(a_1=b_1\) and \(a_2=b_2\). From this, justify the statement that the general solution of (A) on \((-\infty,\infty)\) is \(y=c_1x+c_2x^2\), where \(c_1\) and \(c_2\) are arbitrary constants.
- For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-2xy'+2y=0,\quad y(0)=k_0,\quad y'(0)=k_1\nonumber \] have a solution? What are the solutions?
- Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem \[x^2y''-2xy'+2y=0,\quad y(x_0)=k_0,\quad y'(x_0)=k_1\nonumber \] has a unique solution on \((-\infty,\infty)\).
44.
- Verify that \(y_1=x^3\) and \(y_2=x^4\) satisfy \[x^2y''-6xy'+12y=0 \tag{A} \] on \((-\infty,\infty)\), and that \(\{y_1,y_2\}\) is a fundamental set of solutions of (A) on \((-\infty,0)\) and \((0,\infty)\).
- Show that \(y\) is a solution of (A) on \((-\infty,\infty)\) if and only if \[y=\left\{\begin{array}{rr} a_1x^3+a_2x^4,&x\ge 0,\\[4pt] b_1x^3+b_2x^4,&x<0, \end{array}\right.\nonumber \] where \(a_1\), \(a_2\), \(b_1\), and \(b_2\) are arbitrary constants.
- For what values of \(k_0\) and \(k_1\) does the initial value problem \[x^2y''-6xy'+12y=0, \quad y(0)=k_0,\quad y'(0)=k_1\nonumber \] have a solution? What are the solutions?
- Show that if \(x_0\ne0\) and \(k_0,k_1\) are arbitrary constants then the initial value problem \[x^2y''-6xy'+12y=0, \quad y(x_0)=k_0,\quad y'(x_0)=k_1 \tag{B} \] has infinitely many solutions on \((-\infty,\infty)\). On what interval does (B) have a unique solution?
