5.1.1: Homogeneous Linear Equations (Exercises)

1.

1. Verify that \(y_1=e^{2x}\) and \(y_2=e^{5x}\) are solutions of \[y''-7y'+10y=0 \tag{A}\] on \((-\infty,\infty)\).
2. Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=c_1e^{2x}+c_2e^{5x}\) is a solution of (A) on \((-\infty,\infty)\).
3. Solve the initial value problem \[y''-7y'+10y=0,\quad y(0)=-1,\quad y'(0)=1.\nonumber \]

2.

1. Verify that \(y_1=e^x\cos x\) and \(y_2=e^x\sin x\) are solutions of \[y''-2y'+2y=0 \tag{A}\] on \((-\infty,\infty)\).
2. Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=c_1e^x\cos x+c_2e^x\sin x\) is a solution of (A) on \((-\infty,\infty)\).
3. Solve the initial value problem \[y''-2y'+2y=0,\quad y(0)=3,\quad y'(0)=-2.\nonumber \]

3.

1. Verify that \(y_1=e^x\) and \(y_2=xe^x\) are solutions of \[y''-2y'+y=0 \tag{A}\] on \((-\infty,\infty)\).
2. Verify that if \(c_1\) and \(c_2\) are arbitrary constants then \(y=e^x(c_1+c_2x)\) is a solution of (A) on \((-\infty,\infty)\).
3. Solve the initial value problem \[y''-2y'+y=0,\quad y(0)=7,\quad y'(0)=4.\nonumber \]

4.

1. Verify that \(y_1=1/(x-1)\) and \(y_2=1/(x+1)\) are solutions of \[(x^2-1)y''+4xy'+2y=0 \tag{A}\] on \((-\infty,-1)\), \((-1,1)\), and \((1,\infty)\). What is the general solution of (A) on each of these intervals?
2. Solve the initial value problem \[(x^2-1)y''+4xy'+2y=0,\quad y(0)=-5,\quad y'(0)=1.\nonumber \] What is the interval of validity of the solution?

In Exercises 5-11 compute the Wronskians of the given sets of functions.

5. \(\{1, e^{x}\}\)

6. \(\{e^{x}, e^{x}\sin x\}\)

7. \(\{x+1, x^{2}+2\}\)

8. \(\{x^{1/2}, x^{-1/3}\}\)

9. \(\{\frac{\sin x}{x},\frac{\cos x}{x}\}\)

10. \(\{x\ln |x|, x^{2}\ln |x|\}\)

11. \(\{e^{x}\cos\sqrt{x}, e^{x}\sin\sqrt{x}\}\)

12. 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 prove 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 \(y\equiv0\).

13. 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 \((a,b)\).

1. 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 \((a,b)\).
2. 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 \((a,b)\).

14. 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. Prove that if \(\{z_1,z_2\}\) is a fundamental set of solutions of (A) on \((a,b)\) then so is \(\{y_1,y_2\}\).

15. 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. Prove that \(\{z_1,z_2\}\) is a fundamental set of solutions of (A) on \((a,b)\) if and only if \(\alpha\gamma-\beta\delta\ne0\).

16. Suppose \(y_1\) is differentiable on an interval \((a,b)\) and \(y_2=ky_1\), where \(k\) is a constant. Prove that the Wronskian of \(\{y_1,y_2\}\) is identically zero on \((a,b)\).

17. Let

\[y_1=x^3\quad\mbox{ and }\quad y_2=\left\{\begin{array}{rl} x^3,&x\ge 0,\\ -x^3,&x<0.\end{array}\right.\nonumber \]

Show that the Wronskian of \(\{y_1,y_2\}\) is defined and identically zero on \((-\infty,\infty)\).

18. 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 5.1.2 to prove that \(y_1\) and \(y_2\) are themselves solutions of (A) on \((a,b)\).

19. 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 \((a,b)\). Prove that \(p_1=p_2\) and \(q_1=q_2\) on \((a,b)\).

20. 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 \((a,b)\). Prove 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 \((a,b)\).

21. If \(y=c_1+c_2x^2\) is a family of solutions of \(xy''-y'=0\), show that the initial value problem

\[xy''-y'=0; \quad y(0)=0,\quad y'(0)=1\nonumber\] has no solution that can be obtained from this family. Why does this not contradict Theorem 5.1.1?

22. If \(y=c_1+c_2x^2\) is a family of solutions of \(xy''-y'=0\), show that the initial value problem

\[xy''-y'=0; \quad y(0)=0,\quad y'(0)=0\nonumber\] has an infinite number of solutions that can be obtained from this family. Why does this not contradict Theorem 5.1.1?

23. a. Find the largest interval for which the IVP \[e^xy''+(\cot x)y'+{3\over x-2}y=0; \quad y(1)=2, \quad y'(1)=0\nonumber\] is guaranteed a unique solution.

b. Find the largest interval for which the IVP \[e^xy''+(\cot x)y'+{3\over x-2}y=0; \quad y(3)=2, \quad y'(3)=0\nonumber\] is guaranteed a unique solution.

24. a. Find the largest interval for which the IVP \[(\ln x)y''+(\sec x)y'+{1\over x-5}y=0; \quad y(4)=2, \quad y'(4)=0\nonumber\] is guaranteed a unique solution.

b. Find the largest interval for which the IVP \[(\ln x)y''+(\sec x)y'+{1\over x-5}y=0; \quad y(3/2)=2, \quad y'(3/2)=0\nonumber\] is guaranteed a unique solution.