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Theory of Higher-Order Linear Differential Equations

Fundamentals Of Differential Equations And Boundary Value Problems ยท 123 exercises

Q1E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

xy'''-3y'+exy=x2-1y-2=1,  y'-2=0,  y''-2=2

3 step solution

Q2E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

y'''-xy=sinxy(π)=0,  y'(π)=11,  y''(π)=3

3 step solution

Q3E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

y'''-y''+x-1y=tanxy5=y'5=y''5=1

3 step solution

Q4E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

xx+1y'''-3xy'+y=0y-12=1,  y'-12=y''-12=0

3 step solution

Q5E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

xx+1y'''-y'+xy=0y12=y'12=-1,  y''12=1

3 step solution

Q6E

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

x2-1y'''+exy=lnxy34=1,  y'34=y''34=0

3 step solution

Q7E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{e3x, e5x, e-x} on (-∞, ∞)

2 step solution

Q8E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x2, x2-1, 5} on (-∞, ∞)

2 step solution

Q9E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{sin2x, cos2x, 1} on  (-∞, ∞)

2 step solution

Q10E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{sinx, cosx, tanx} on (-π2, π2)

2 step solution

Q11E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x-1, x12, x} on (0, ∞)

2 step solution

Q12E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{cos2x, cos2x, sin2x} on (-∞, ∞)

2 step solution

Q13E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x, x2, x3, x4} on (-∞, ∞)

2 step solution

Q14E

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x, xex, 1} on (-∞, ∞)

2 step solution

Q15E

Using the Wronskian in this Problem, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.

y'''+2y''-11y'-12y=0;{e3x, e-x, e-4x}

2 step solution

Q16E

Using the Wronskian in this Problem, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.

y'''-y''+4y'-4y=0;{ex, cos2x, sin2x}

2 step solution

Q17E

Using the Wronskian in this Problem, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.

x3y'''-3x2y''+6xy'-6y=0,  x>0;{x, x2, x3}

2 step solution

Q18E

Using the Wronskian in this Problem, verify that the given functions form a fundamental solution set for the given differential equation and find a general solution.

y4-y=0;{ex, e-x, cosx, sinx}

2 step solution

Q19E

A particular solution and a fundamental solution set are given for a nonhomogeneous equation and its corresponding homogeneous equation. 

(a) Find a general solution to the non-homogeneous equation. 

(b) Find the solution that satisfies the specified initial condition.

y'''+y''+3y'-5y=2+6x-5x2;y(0)=-1,    y'(0)=1,    y''(0)=-3;yp=x2;    {ex,  e-xcos2x,  e-xsin2x}

7 step solution

Q20E

A particular solution and a fundamental solution set are given for a nonhomogeneous equation and its corresponding homogeneous equation. 

(a) Find a general solution to the nonhomogeneous equation. 

(b) Find the solution that satisfies the specified initial condition.

xy'''-y''=-2;y(1)=2,    y'(1)=-1,    y''(1)=-4;yp=x2;    {1,  x,  x3}

7 step solution

Q21E

A particular solution and a fundamental solution set are given for a nonhomogeneous equation and its corresponding homogeneous equation. 

(a) Find a general solution to the nonhomogeneous equation. 

(b) Find the solution that satisfies the specified initial condition.

x3y'''+xy'-y=3-lnx,          x>0;y(1)=3,    y'(1)=3,    y''(1)=0;yp=lnx;    {x,  xlnx,  xlnx2}

6 step solution

Q23E

Let\({\bf{L}}\left[{\bf{y}}\right]{\bf{=y'''+y'+ xy,}}\,\,\,\,\,\,\,{{\bf{y}}_{\bf{1}}}\left({\bf{x}}\right){\bf{=sinx,}}\)and\({{\bf{y}}_{\bf{2}}}\left({\bf{x}}\right){\bf{=x}}\).Verifythat\({\bf{L}}\left[{{{\bf{y}}_{\bf{1}}}}\right]\left( {\bf{x}} \right){\bf{=xsinx,}}\)and\({\bf{L}}\left[ {{{\bf{y}}_{\bf{2}}}} \right]\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 1}}\). Then use the superposition principle (linearity) to find a solution to the differential equation: 

 

(a) \({\bf{L}}\left[ {\bf{y}} \right]{\bf{ = 2xsinx - }}{{\bf{x}}^{\bf{2}}}{\bf{ - 1}}\)

(b) \({\bf{L}}\left[ {\bf{y}} \right]{\bf{ = 4}}{{\bf{x}}^{\bf{2}}}{\bf{ + 4 - 6xsinx}}\)

4 step solution

Q24E

Let \({\bf{L}}\left[ {\bf{y}} \right]{\bf{ = y''' - xy'' + 4y' - 3xy,}}\,\,\,\,\,\,\,{{\bf{y}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = cos2x,}}\)and\({{\bf{y}}_{\bf{2}}}\left( {\bf{x}} \right){\bf{ =  - }}\frac{{\bf{1}}}{{\bf{3}}}\). Verify that \({\bf{L}}\left[ {{{\bf{y}}_{\bf{1}}}} \right]\left( {\bf{x}} \right){\bf{ = xcos2x,}}\)and\({\bf{L}}\left[ {{{\bf{y}}_{\bf{2}}}} \right]\left( {\bf{x}} \right){\bf{ = x}}\). Then use the superposition principle (linearity) to find a solution to the differential equation: 

 

(a) \({\bf{L}}\left[ {\bf{y}} \right]{\bf{ = 7xcos2x - 3x}}\)

(b) \({\bf{L}}\left[ {\bf{y}} \right]{\bf{ =  - 6xcos2x + 11x}}\)

 

4 step solution

Q25E

Prove that L defined in (7) is a linear operator by verifying that properties (9) and (10) hold for any n-times differentiable functions \({\bf{y,}}\,{{\bf{y}}_{\bf{1}}}{\bf{,}}\,...{\bf{,}}\,{{\bf{y}}_{\bf{m}}}\)  on (a, b).

 

3 step solution

Q26E

Existence of Fundamental Solution Sets. By Theorem 1, for each j = 1, 2, . . ., n there is a unique solution \({{\bf{y}}_{\bf{j}}}\left( {\bf{x}} \right)\) to equation (17) satisfying the initial conditions 

\({{\bf{y}}_{\bf{j}}}^{\left( {\bf{k}} \right)}\left( {{{\bf{x}}_{\bf{0}}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{1,}}\,\,\,\,\,{\bf{for}}\,{\bf{k = j - 1,}}\\{\bf{0,}}\,\,\,\,{\bf{for}}\,{\bf{k}} \ne {\bf{j - 1,}}\,\,{\bf{0}} \le {\bf{k}} \le {\bf{n - 1}}\end{array} \right\}\)

(a) Show that \(\left\{ {{{\bf{y}}_{\bf{1}}}{\bf{,}}{{\bf{y}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{y}}_{\bf{n}}}} \right\}\) is a fundamental solution set for (17). [Hint: Write out the Wronskian at\({{\bf{x}}_{\bf{0}}}\)].

(b) For given initial values\({\gamma _{\bf{0}}}{\bf{,}}{\gamma _{\bf{1}}}{\bf{,}}{\gamma _{\bf{2}}}{\bf{,}}...{\bf{,}}{\gamma _{{\bf{n - 1}}}}\), express the solution y(x) to (17) satisfying \({{\bf{y}}^{\left( {\bf{k}} \right)}}\left( {{{\bf{x}}_{\bf{0}}}} \right){\bf{ = }}{\gamma _{\bf{k}}},\,\,{\bf{k = 0,1,}}...{\bf{,n - 1}}\),[as in equations (4)] in terms of this fundamental solution set.

3 step solution

Q27E

Show that the set of functions\(\left\{ {{\bf{1,}}\,{\bf{x,}}\,{{\bf{x}}^{\bf{2}}}{\bf{,}}...{\bf{,}}\,{{\bf{x}}^{\bf{n}}}} \right\}\), where n is a positive integer, is linearly independent on every open interval (a, b). 

[Hint: Use the fact that a polynomial of degree at most n has no more than n zeros unless it is identically zero.]

 

2 step solution

Q28E

Show that the set of functions\(\left\{ {{\bf{1,}}\,{\bf{cosx,}}\,{\bf{sinx,}}...{\bf{,}}\,{\bf{cosnx,}}\,{\bf{sinnx}}} \right\}\), 

where n is a positive integer is linearly independent on every open interval (a, b). Prove this in the special case n = 2 and (a, b) = \(\left( {{\bf{ - }}\infty {\bf{,}}\,\infty } \right)\)

 

3 step solution

Q29E

(a) Show that if \({{\bf{f}}_{\bf{1}}}{\bf{,}}\,{{\bf{f}}_{\bf{2}}}{\bf{,}}\,...{\bf{,}}\,{{\bf{f}}_{\bf{m}}}\)are linearly independent on (-1, 1), then they are linearly independent on\(\left( {{\bf{ - }}\infty {\bf{,}}\,\infty } \right)\). 

(b) Give an example to show that if \({{\bf{f}}_{\bf{1}}}{\bf{,}}\,{{\bf{f}}_{\bf{2}}}{\bf{,}}\,...{\bf{,}}\,{{\bf{f}}_{\bf{m}}}\) are linearly independent on\(\left( {{\bf{ - }}\infty {\bf{,}}\,\infty } \right)\), then they need not be linearly independent on (-1, 1).

2 step solution

Q30E

To prove Abel’s identity (26) for n = 3, proceed as follows: 

 

(a) Let\({\bf{W}}\left( {\bf{x}} \right){\bf{ = W}}\left[ {{{\bf{y}}_{\bf{1}}}{\bf{,}}\,{{\bf{y}}_{\bf{2}}}{\bf{,}}\,{{\bf{y}}_{\bf{3}}}} \right]\left( {\bf{x}} \right)\). Use the product rule for differentiation to show

\({\bf{W'}}\left( {\bf{x}} \right){\bf{ = }}\left| {\begin{array}{*{20}{c}}{{{\bf{y}}_{\bf{1}}}{\bf{'}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{'}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{''}}}&{{{\bf{y}}_{\bf{2}}}{\bf{''}}}&{{{\bf{y}}_{\bf{3}}}{\bf{''}}}\end{array}} \right|{\bf{ + }}\left| {\begin{array}{*{20}{c}}{{{\bf{y}}_{\bf{1}}}}&{{{\bf{y}}_{\bf{2}}}}&{{{\bf{y}}_{\bf{3}}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{''}}}&{{{\bf{y}}_{\bf{2}}}{\bf{''}}}&{{{\bf{y}}_{\bf{3}}}{\bf{''}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{''}}}&{{{\bf{y}}_{\bf{2}}}{\bf{''}}}&{{{\bf{y}}_{\bf{3}}}{\bf{''}}}\end{array}} \right|{\bf{ + }}\left| {\begin{array}{*{20}{c}}{{{\bf{y}}_{\bf{1}}}}&{{{\bf{y}}_{\bf{2}}}}&{{{\bf{y}}_{\bf{3}}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{'}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{'''}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'''}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'''}}}\end{array}} \right|\)

(b) Show that the above expression reduces to

 

(32) \({\bf{W'}}\left( {\bf{x}} \right){\bf{ = }}\left| {\begin{array}{*{20}{c}}{{{\bf{y}}_{\bf{1}}}}&{{{\bf{y}}_{\bf{2}}}}&{{{\bf{y}}_{\bf{3}}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{'}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'}}}\\{{{\bf{y}}_{\bf{1}}}{\bf{'''}}}&{{{\bf{y}}_{\bf{2}}}{\bf{'''}}}&{{{\bf{y}}_{\bf{3}}}{\bf{'''}}}\end{array}} \right|\)

(c) Since each \({{\bf{y}}_{\bf{i}}}\) satisfies (17), show that

(33) \({{\bf{y}}_{\bf{i}}}^{\left( {\bf{3}} \right)}\left( {\bf{x}} \right){\bf{ =   - }}\sum\limits_{{\bf{k = 1}}}^{\bf{3}} {{{\bf{p}}_{\bf{k}}}\left( {\bf{x}} \right){{\bf{y}}_{\bf{i}}}^{\left( {{\bf{3 - k}}} \right)}\left( {\bf{x}} \right),\,\,\,\,\,\,\,\,\,\,\left( {{\bf{i = 1,2,3}}} \right)} \)

(d) Substituting the expressions in (33) into (32), show that (34) \({\bf{W'}}\left( {\bf{x}} \right){\bf{ =  - }}{{\bf{p}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{W}}\left( {\bf{x}} \right)\)

(e) Deduce Abel’s identity by solving the first-order differential equation (34).

5 step solution

Q31E

Reduction of Order. If a nontrivial solution f(x) is known for the homogeneous equation

,y(n)+p1(x)y(n-1)+...+pn(x)y=0

the substitution y(x)=v(x)f(x) can be used to reduce the order of the equation for second-order equations. By completing the following steps, demonstrate the method for the third-order equation

(35) y'''-2y''-5y'+6y=0

given that f(x)=ex is a solution.

(a) Set y(x)=v(x)ex and compute y′, y″, and yโ€ด. 

(b) Substitute your expressions from (a) into (35) to obtain a second-order equation in. w=v'

(c) Solve the second-order equation in part (b) for w and integrate to find v. Determine two linearly independent choices for v, say, v1and v2.

(d) By part (c), the functions y1(x)=v1(x)ex and y2(x)=v2(x)ex are two solutions to (35). Verify that the three solutions ex, y1(x), and y2(x) are linearly independent on (-∞, ∞)

5 step solution

Q32E

Given that the function f(x)=x  is a solution to y'''-x2y'+xy=0, show that the substitution y(x)=v(x)f(x)=v(x)x reduces this equation to, xw''+3w'-x3w=0 where w=v'.

2 step solution

Q33E

Use the reduction of order method described in Problem 31 to find three linearly independent solutions to, given y'''-2y''+y'-2y=0 that f(x)=e2x is a solution.

6 step solution

Q34E

Constructing Differential Equations. Given three functions that f1(x), f2(x),f3(x) are each three times differentiable and whose Wronskian is never zero on (a, b), show that the equation 

 

|f1(x)f2(x)f3(x)yf1'(x)f2'(x)f3'(x)y'f1''(x)f2''(x)f3''(x)y''f1'''(x)f2'''(x)f3''(x)y'''|=0

 

is a third-order linear differential equation for which {f1, f2 f3} is a fundamental solution set. What is the coefficient of yโ€ด in this equation?

2 step solution

Q35E

Use the result of Problem 34 to construct a third-order differential equation for which {x, sinx, cosx}  is a fundamental solution set.

2 step solution

Q1E

Find a general solution for the differential equation with x as the independent variable.

y'''+2y''-8y'=0 

2 step solution

Q2E

Find a general solution for the differential equation with x as the independent variable.

y'''-3y''-y'+3y=0

2 step solution

Q3E

Find a general solution for the differential equation with x as the independent variable.6z'''+7z''-z'-2z=0

2 step solution

Q4E

Find a general solution for the differential equation with x as the independent variable:y'''−3y''−y'+3y=0

3 step solution

Q6E

Find a general solution for the differential equation with x as the independent variable:

 y'''−y''+2y=0

3 step solution

Q7E

Find a general solution for the differential equation with x as the independent variable:

2y'''−y''−10y'−7y=0

3 step solution

Q8E

Find a general solution for the differential equation with x as the independent variable:

 2y'''+5y''−13y'+7y=0

2 step solution

Q9E

Find a general solution for the differential equation with x as the independent variable:

u'''−9u''+27u'−27u=0

2 step solution

Q11E

Find a general solution for the differential equation with x as the independent variable:

 y(4)+4y'''+6y''+4y'+y=0

2 step solution

Q12E

Find a general solution for the differential equation with x as the independent variable:

 y'''+5y''+3y'−9y=0

2 step solution

Q13E

 Find a general solution for the differential equation with x as the independent variable:

 y(4)+4y''+4y=0

2 step solution

Q14E

Find a general solution for the differential equation with x as the independent variable:

 y(4)+2y'''+10y''+18y'+9y=0

2 step solution

Q15E

Find a general solution to the givenhomogeneous equation. (D−1)2(D+3)(D2+2D+5)2[y]=0

3 step solution

Q16E

Find a general solution to the givenhomogeneous equation.  (D+1)2(D−6)3(D+5)(D2+1)(D2+4)2[y]=0

3 step solution

Q17E

Find a general solution to the givenhomogeneous equation.(D+4)(D−3)(D+2)3(D2+4D+5)2D5[y]=0


2 step solution

Q18E

Find a general solution to the givenhomogeneous equation.

 (D−1)3(D−2)(D2+D+1)(D2+6D+10)3[y]=0

2 step solution

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Theory of Higher-Order Linear Differential Equations - Fundamentals Of Differential Equations And Boundary Value Problems Solutions | StudyQuestionHub