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Mathematics LibreTexts

Section 13.1 Answers

  • Page ID
    30089
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    2. \(y=-x+\frac{2}{e-1}(e^{x}-e^{(x-1)})\)

    3. \(y=x^{2}-\frac{x^{3}}{3}+cx\) with \(c\) arbitrary 

    4. \(y=-x+2e^{x}+e^{-(x-1)}\)

    5. \(y=\frac{1}{4}+\frac{11}{4}\cos 2x+\frac{9}{4}\sin 2x\)

    6. \(y=(x^{2}+13-8x)e^{x}\)

    7. \(y=2e^{2x}+\frac{3(5e^{3x}-4e^{4x})}{e^15-16e)}+\frac{2e^{4}(4e^{3(x-1)}-3e^{4(x-1)})}{16e-15}\)

    8. \(\int_{a}^{b}tF(t)dt=0\quad y=-x\int_{x}^{a}F(t)dt-\int_{0}^{x}tF(t)dt+c_{1}x\) with \(c_{1}\) arbitrary

    9. 

    1. \(b-a\neq k\pi \) (\(k=\) integer) \(y=\frac{\sin (x-1)}{\sin (b-a)}\int_{x}^{b} F(t)\sin (t-b)dt+\frac{\sin (x-b)}{\sin (b-a)}\int_{a}^{x} F(t)\sin (t-a)dt\)
    2. \(\int_{a}^{b}F(t)\sin (t-a)dt=0\quad y=-\sin (x-a)\int_{x}^{b} F(t)\cos (t-a)dt-\cos (x-a)\int_{a}^{x}F(t)\sin (t-a)dt+c_{1}\sin (x-a)\text{ with } c_{1}\text{ arbitrary}\)

    10. 

    1. \(b-a\neq (k+1/2)\pi \: (k=\text{ integer})\quad y=-\frac{\sin (x-a)}{\cos (b-a)}\int_{x}^{b}F(t)\cos (t-b)dt-\frac{\cos (x-b)}{\cos (b-a)}\int_{x}^{b}F(t)\sin (t-a)dt\)
    2. \(\int_{a}^{b}F(t)\sin (t-a)dt=0\quad y=-\sin (x-a)\int_{x}^{b}F(t)\cos (t-a)dt-\cos (x-a)\int_{a}^{x}F(t)\sin (t-a)dt+c_{1}\sin (x-a)\text{ with }c_{1}\text{ arbitrary}\)

    11. 

    1. \(b-a\neq k\pi (k=\text{ integer})\quad y=\frac{\cos (x-a)}{\sin (b-a)}\int_{x}^{b}F(t)\cos (t-b)dt+\frac{\cos (x-b)}{\sin (b-a)}\int_{a}^{x}F(t)\cos (t-a)dt\)
    2. \(\int_{a}^{b}F(t)\cos (t-a)dt =0\quad y=\cos (x-a)\int_{x}^{b}F(t)\sin (t-a)dt +\sin (x-a)\int_{a}^{x}F(t)\cos (t-a)dt+c_{1}\cos (x-a)\text{ with }c_{1}\text{ arbitrary}\)

    12. \(y=\frac{\sinh (x-a)}{\sinh (b-a)}\int_{x}^{b}F(t)\sinh (t-b)dt+\frac{\sinh (x-b)}{\sinh (b-a)}\int_{a}^{x}F(t)\sinh (t-a)dt\)

    13. \(y=-\frac{\sinh (x-a)}{\cosh (b-a)}\int_{x}^{b}F(t)\cosh (t-b)dt-\frac{\cosh (x-b)}{\cosh (b-a)}\int_{a}^{x}F(t)\sinh (t-a)dt\)

    14. \(y=-\frac{\cosh (x-a)}{\sinh (b-a)}\int_{x}^{b}F(t)\cosh (t-b)dt-\frac{\cosh (x-b)}{\sinh (b-a)}\int_{a}^{x}F(t)\cosh (t-a)dt\)

    15. \(y=-\frac{1}{2}\left(e^{x}\int_{x}^{b}e^{-t}F(t)dt+e^{-x}\int_{a}^{x}e^{t}F(t)dt\right)\)

    16. If \(\omega\) isn't a positive integer, then \(y=\frac{1}{\omega\sin\omega\pi}\left(\sin\omega x\int_{x}^{\pi}F(t)\sin\omega (t-\pi )dt+\sin\omega (x-\pi )\int_{0}^{x}F(t)\sin\omega tdt\right).\) If \(\omega =n\) (positive integer), then \(\int_{0}^{\pi }F(t)\sin ntdt=0\) is necessary for existence of a solution. In this case, \(y=-\frac{1}{n}\left(\sin nx\int_{x}^{\pi}F(t)\cos ntdt+\cos nx\int_{0}^{x}F(t)\sin ntdt\right)+c_{1}\sin nx\) with \(c_{1}\)arbitrary.

    17. If \(\omega\neq n+1/2 (n=\) integer), then \(y=-\frac{\sin\omega x}{\omega\cos\omega\pi }\int_{x}^{\pi}F(t)\cos\omega (t-\pi )dt-\frac{\cos\omega (x-\pi )}{\omega\cos\omega\pi}\in_{0}^{x}F(t)\sin\omega tdt\). If \(\omega = n+1/2 (n=\) integer), then \(\int_{0}^{\pi}F(t)\sin (n+1/2)tdt=0\) is necessary for existence of a solution. In this case, \(y=-\frac{\sin (n+1/2)x}{n+1/2}\int_{x}^{\pi}F(t)\cos (n+1/2)tdt-\frac{\cos (n+1/2)x}{n+1/2}\int_{0}^{x}F(t)\sin (n+1/2)tdt+c_{1}\sin (n+1/2)x\) with \(c_{1}\) arbitrary.

    18. If \(\omega\neq n+1/2\: (n=\) integer), then \(y=\frac{\cos\omega x}{\omega\cos\omega\pi}\int_{x}^{\pi}F(t)\sin\omega (t-\pi )dt+\frac{\sin\omega (x-\pi )}{\omega\cos\omega\pi}\int_{0}^{x}F(t)\cos\omega tdt\). If \(\omega = n+1/2 (n=\) integer), then \(\int_{0}^{\pi }F(t)\cos (n+1/2)tdt=0\) is necessary for existence of a solution. In this case, \(y=\frac{\cos (n+1/2)x}{n+1/2}\int_{x}^{\pi}F(t)\sin (n+1/2)tdt+\frac{\sin (n+1/2)x}{n+1/2}\int_{0}^{x}F(t)\cos (n+1/2)tdt+c_{1}\cos (n+1/2)x\) with \(c_{1}\) arbitrary. 

    19. If \(\omega\) isn't a positive integer, then \(y=\frac{1}{\omega\sin\omega\pi}\left(\cos\omega x\int_{x}^{\pi }F(t)\cos\omega (t-\pi )dt+\cos\omega (x-\pi )\int_{0}^{x}F(t)\cos\omega tdt\right)\). If \(\omega = n\) (positive integer), then \(\int_{0}^{\pi }F(t)\cos ntdt=0\) is necessary for existence of a solution. In this case, \(y=-\frac{1}{n}\left(\cos nx\int_{x}^{\pi}F(t)\sin ntdt+\sin nx\int_{0}^{x}F(t)\cos ntdt\right) +c_{1}\cos nx\) with \(c_{1}\) arbitrary.

    20. \(y_{1}=B_{1}(z_{2})z_{1}-B_{1}(z_{1})z_{2}\)

    21. 

    1. \(G(x,t)=\left\{\begin{array}{cl}{\frac{(t-a)(x-b)}{b-a}}&{a\leq t\leq x,}\\{\frac{(x-a)(t-b)}{(b-a)}}&{x\leq t\leq b}\end{array} \right.\quad y=\frac{1}{b-a}\left( (x-a)\int_{x}^{b}(t-b)F(t)dt+(x-b)\int_{a}^{x}(t-a)F(t)dt\right)\)
    2. \(G(x,t)=\left\{\begin{array}{cl}{a-t}&{a\leq t\leq x}\\{a-x}&{x\leq t\leq b}\end{array} \right.\quad y=(a-x)\int_{x}^{b}F(t)dt+\int_{a}^{x}(a-t)F(t)dt\)
    3. \(G(x,t)=\left\{\begin{array}{cl}{x-b}&{a\leq t\leq x}\\{t-b}&{x\leq t\leq b}\end{array}\right.\quad y=\int_{x}^{b}(t-b)F(t)dt+(x-b)\int_{a}^{x}F(t)dt\)
    4. \(\int_{a}^{b}F(t)dt=0\) is a necessary condition for existence of a solution. Then \(y=\int_{x}^{b}tF(t)dt+x\int_{a}^{x}F(t)dt+c_{1}\) with \(c_{1}\) arbitrary.

    22. \(G(x,t)=\left\{\begin{array}{cl}{-\frac{(2+t)(3-x)}{5},}&{0\leq t\leq x,}\\{-\frac{(2+x)(3-t)}{5},}&{x\leq t\leq 1}\end{array}\right.\)

    1. \(y=\frac{x^{2}-x-2}{2}\)
    2. \(y=\frac{5x^{2}-7x-14}{30}\)
    3. \(y=\frac{5x^{4}-9x-18}{60}\)

    23. \(G(x,t)=\left\{\begin{array}{cl}{\frac{\cos t\sin x}{t^{3/2}\sqrt{x}},}&{\frac{\pi }{2}\leq t\leq x,}\\{\frac{\cos x\sin t}{t^{3/2}\sqrt{x}},}&{x\leq t\leq \pi}\end{array}\right.\)

    1. \(y=\frac{1+\cos x-\sin x}{\sqrt{x}}\)
    2. \(y=\frac{x+\pi\cos x-\pi /2\sin x}{\sqrt{x}}\)

    24.  \(G(x,t)=\left\{\begin{array}{cl}{\frac{(t-1)x(x-2)}{t^{3}},}&{1\leq t\leq x,}\\{\frac{x(x-1)(t-2)}{t^{3}},}&{x\leq t\leq 2}\end{array}\right.\)

    1. \(y=x(x-1)(x-2)\)
    2. \(y=x(x-1)(x-2)(x+3)\)

    25.  \(G(x,t)=\left\{\begin{array}{cl}{-\frac{1}{22}\left(3+\frac{1}{t^{2}}\right)\left(x+\frac{4}{x}\right),}&{1\leq x\leq 2,}\\{-\frac{1}{22}\left(3x+\frac{1}{x}\right)\left(1+\frac{4}{t^{2}}\right),}&{x\leq t\leq 2}\end{array}\right.\)

    1. \(y=\frac{x^{2}-11x+4}{11x}\)
    2. \(y=\frac{11x^{3}-45x^{2}-4}{33x}\)
    3. \(y=\frac{11x^{4}-139x^{2}-28}{88x}\)

    26. \(\alpha (\rho +\delta )-\beta\rho\neq 0\quad  G(x,t)=\left\{\begin{array}{cl}{\frac{(\beta -\alpha t)(\rho +\delta -\rho x)}{\alpha (\rho +\delta )-\beta\rho },}&{0\leq t\leq x,}\\{\frac{(\beta -\alpha x)(\rho +\delta -\rho t)}{\alpha (\rho +\delta )-\beta\rho },}&{x\leq t\leq 1}\end{array}\right.\)

    27. \(\alpha\delta -\beta\rho\neq 0\quad G(x,t)=\left\{\begin{array}{cl}{\frac{(\beta\cos t-\alpha\sin t)(\delta\cos x-\rho\sin x)}{\alpha\delta -\beta\rho },}&{0\leq t\leq x,}\\{\frac{(\beta\cos x-\alpha\sin x)(\delta\cos t-\rho\sin x)}{\alpha\delta -\beta\rho }}&{x\leq t\leq\pi }\end{array}\right.\)

    28. \(\alpha\rho +\beta\delta\neq 0\quad G(x,t)=\left\{\begin{array}{cl}{\frac{(\beta\cos t-\alpha\sin t)(\rho\cos x+\delta\sin x)}{\alpha\rho +\beta\delta },}&{x\leq t\leq\pi }\\{\frac{(\beta\cos x-\alpha\sin x)(\rho\cos t+\delta\sin t)}{\alpha\rho +\beta\delta }}&{0\leq t\leq x}\end{array}\right.\)

    29. \(\alpha\delta -\beta\rho\neq 0\quad G(x,t)=\left\{\begin{array}{cl}{\frac{e^{(x-t)}(\beta\cos t-(\alpha +\beta )\sin t)(\delta\cos x-(\rho +\delta )\sin x) }{\alpha\delta -\beta\rho }}&{0\leq t\leq x,}\\{\frac{e^{x-t}(\beta\cos x-(\alpha +\beta )\sin x)(\delta\cos t-(\rho +\delta )\sin t)}{\alpha\delta -\beta\rho }}&{x\leq t\leq\pi}\end{array}\right.\)

    30. \(\beta\delta +(\alpha +\beta )\neq 0\quad G(x,t)=\left\{\begin{array}{cl}{\frac{e^{x-t}(\beta\cos t-(\alpha +\beta )\sin t)((\rho +\delta )\cos x+\delta\sin x)}{\beta\delta + (\alpha +\beta )(\rho +\delta )},}&{0\leq t\leq x,}\\{\frac{e^{x-t}(\beta\cos x-(\alpha +\beta )\sin x)((\rho +\delta )\cos t+\delta\sin t)}{\beta\delta +(\alpha +\beta )(\rho +\delta )},}&{x\leq t\leq \pi /2}\end{array}\right.\)

    31. \((\rho +\delta )(\alpha -\beta )e^{(b-a)}-(\rho -\delta )(\alpha +\beta )e^{(a-b)}\neq 0\quad G(x,t)=\left\{\begin{array}{cl}{\frac{((\alpha -\beta )e^{(t-a)}-(\alpha +\beta )e^{-(t-a)}((\rho -\delta )e^{(x-b)}-(\rho +\delta )e^{-(x-b)} )}{2\left[ (\rho +\delta )(\alpha -\beta )e^{(b-a)}-(\rho -\delta )(\alpha +\beta )e^{(a-b)}\right] },}&{0\leq t\leq x,}\\{\frac{((\alpha -\beta )e^{(x-a)}-(\alpha +\beta )e^{-(x-a)})((\rho -\delta )e^{(t-b)}-(\rho +\delta )e^{-(t-b)} )}{2\left[ (\rho +\delta )(\alpha -\beta )e^{(b-a)} -(\rho -\delta )(\alpha +\beta )e^{(a-b)}\right] },}&{x\leq t\leq\pi}\end{array}\right.\)