Ejercicios de repaso Capítulo 1

En los ejercicios 1 a 20, Compruebe si la función indicada es una solución o no de la E.D. dada.

  1. $\hspace{0.5cm}y=e^\frac{x}{2}$

de $\hspace{0.3cm}2y^{'}+y=0$

  1. $\hspace{0.5cm}y=e^{3x}+10e^{2x}$

de $\hspace{0.3cm}\dfrac{dy}{dx}-2y=e^{3x}$

  1. $\hspace{0.5cm}y=x\ln \left| x\right|$

de $\hspace{0.3cm}y^{'}-\dfrac{1}{x}y=1$

  1. $\hspace{0.5cm}y=5tan5x$

de $\hspace{0.3cm}y^{'}=25+y^2$

  1. $\hspace{0.5cm}y=\dfrac{senx}{3x}$

de $\hspace{0.3cm}xy^{'}+y=cosx$

  1. $\hspace{0.5cm}y=\dfrac{-3}{3x+2}$

de $\hspace{0.3cm}y^{'}=3y^2$

  1. $\hspace{0.5cm}y-\dfrac{1}{cosx}=0$

de $\hspace{0.3cm}y^{'}-ytanx=0$

  1. $\hspace{0.5cm}y=\dfrac{x}{cosx}$

de $\hspace{0.2cm}xy^{'}-y=xtanxsecx$

  1. $\hspace{0.5cm}y=1+c\sqrt {1-x^{2}}$

de $\hspace{0.2cm}(1-x^2)y^{'}+xy=x$

  1. $\hspace{0.5cm}y=e^{x}-e^{-x}$

de $\hspace{0.2cm}y^{'}=y+2e^{-x}$

  1. $\hspace{0.5cm}y=e^{x}cosx$

de $\hspace{0.2cm}y^{''}-2y^{'}+2y=0$

  1. $\hspace{0.5cm}y=xe^{x}$

de $\hspace{0.2cm}2y^{''}-3y^{'}+y=0$

  1. $\hspace{0.5cm}y=x-\ln \left| x\right|$

de $\hspace{0.01cm}x^2y^{''}+xy^{'}-y=\ln \left| x\right|$

  1. $\hspace{0.5cm}y=e^{3x}cos{2x}$

de $\hspace{0.2cm}y^{''}-6y^{'}+13y=0$

  1. $\hspace{0.5cm}y=e^{-x}cos\frac{1}{2}x$

de $\hspace{0.2cm}4y^{''}+8y^{'}+5y=0$

  1. $\hspace{0.5cm}y=2x^\frac{1}{2}-x^\frac{1}{2}\ln \left| x\right|$

de $\hspace{0.2cm}4x^2y^{''}+y=0$

  1. $\hspace{0.5cm}y=e^{x}(3cos{2x}+sen{2x})$

de $\hspace{0.2cm}y^{''}-2y^{'}+5y=0$

  1. $\hspace{0.5cm}y=-(cosx)\ln \left| secx+tanx\right|$

de $\hspace{0.2cm}y^{''}+y=tanx$

  1. $\hspace{0.5cm}y=cosx\ln \left| cosx\right|+xsenx$

de $\hspace{0.2cm}y^{''}+y=secx$

  1. $\hspace{0.5cm}y=xcos(\ln \left| x\right|)$

de $\hspace{0.2cm}x^2y^{''}-xy^{'}+2y=0$

En los ejercicios 21 a 27, Demuestre que la función indicada es una solución implicita de la E.D. dada.

  1. $\hspace{0.5cm}c=\dfrac{5x^2}{2}+4xy-2y^4$

de $\hspace{0.2cm}(4x-8y^3)\dfrac{dy}{dx}=-5x-4y$

  1. $\hspace{0.5cm}c=x^2y^2-3x+4y$

de $\hspace{0.2cm}(2y^2x-3)=-(2yx^2+4)y^{'}$

  1. $\hspace{0.5cm}c=\dfrac{x^4}{4}+xy^3$

de $\hspace{0.2cm}(x^3+y^3)dx+3xy^2dy=0$

  1. $\hspace{0.5cm}\ln \left| y\right|+y^2=-cosx+c$

de $\hspace{0.2cm}ysenxdx=(1+2y^2)dy$

  1. $\hspace{0.5cm}y^2-x^3+8=0$

de $\hspace{0.2cm}\dfrac{dy}{dx}-\dfrac{3x^2}{2y}=0$

  1. $\hspace{0.5cm}x^2y^3-3x+2y=c$

de $\hspace{0.05cm}(3x^2y^2+2)dy=-(2xy^3-3)dx$

  1. $\hspace{0.5cm}y-\dfrac{1}{y}=tan^{-1}x+c$

de $\hspace{0.2cm}(1+x^2+y^2+x^2y^2)dy=y^2dx$

En los ejercicios 28 a 38, Se le da una ED de primer orden, su solución y una condición inicial, determinar el valor de la constante.

  1. $\hspace{0.2cm}y^{'}=y-y^2$

$y=\dfrac{1}{1+ce^{-x}}$

$y(0)=-\dfrac{1}{3}$

  1. $\hspace{0.2cm}y^{'}=y-y^2$

$y=\dfrac{1}{1+ce^{-x}}$

$y(-1)=2$

  1. $\hspace{0.2cm}y^{'}+2xy^2=0$

$y=\dfrac{1}{x^{2}+c}$

$y\left( \dfrac {1}{2}\right) =-4$

  1. $\hspace{0.2cm}y^{'}+2xy^2=0$

$y=\dfrac{1}{x^{2}+c}$

$y(0)=1$

  1. $\hspace{0.2cm}y^{'}+2xy^2=0$

$y=\dfrac{1}{x^{2}+c}$

$y(-2)=\dfrac{1}{2}$

  1. $\hspace{0.2cm}yy^{'}+6x=0$

$y^2=-6x^2+c$

$y(0)=4$

  1. $\hspace{0.2cm}y^{'}=1+y^2$

$y=tanx(x+c)$

$y\left( \dfrac {\pi}{4}\right) =1$

  1. $\hspace{0.2cm}yy^{'}=e^{2x}+1$

$y^2=e^{2x}+2x+c$

$y(0)=\dfrac{1}{2}$

  1. $\hspace{0.2cm}y^2y^{'}-4x=0$

$y^3=6x^2+c$

$y\left( \dfrac {1}{2}\right) =0$

  1. $\hspace{0.2cm}y^{'}+y=0$

$y=ce^{-x}$

$y(0)=2$

  1. $\hspace{0.2cm}y^{'}+3x^2y=0$

$y=ce^{-x^3}$

$y(0)=7$

En los ejercicios 39 a 47, Se le da una ED de segundo orden, su solución y las condiciones iniciales, determinar el valor de las constantes.

  1. $\hspace{0.2cm}x^{''}+x=0$

    $x=c_1cost+c_2sent$

      $x(0)=-1$

        $x^{'}(0)=8$

          1. $\hspace{0.2cm}x^{''}+x=0$

            $x=c_1cost+c_2sent$

              $x\left( \dfrac {\pi}{2}\right) =0$

                $x^{'}\left( \dfrac {\pi}{2}\right) =1$

                  1. $\hspace{0.2cm}x^{''}+x=0$

                    $x=c_1cost+c_2sent$

                      $x\left( \dfrac {\pi}{6}\right) =\dfrac{1}{2}$

                        $x^{'}\left( \dfrac {\pi}{6}\right) =0$

                          1. $\hspace{0.2cm}y^{''}-y=0$

                            $y=c_1e^x+c_2e^{-x}$

                              $y(0)=1$

                                $y^{'}(0)=2$

                                  1. $\hspace{0.2cm}y^{''}-y=0$

                                    $y=c_1e^x+c_2e^{-x}$

                                      $y(-1)=5$

                                        $y^{'}(-1)=-5$

                                          1. $\hspace{0.2cm}2y^{''}+y^{'}-y=0$

                                            $y=c_1e^{\frac{x}{2}}+c_2e^{-x}$

                                              $y(0)=0$

                                                $y^{'}(0) =1$

                                                  1. $\hspace{0.2cm}y^{''}+y=cosx+4$

                                                    $y=c_1xsenx+c_2$

                                                      $y(0)=4$

                                                        $y^{'}\left( \dfrac {\pi}{2}\right)=1$

                                                          1. $\hspace{0.2cm}y^{''}-3y^{'}-4y=0$

                                                            $y=c_1e^{4x}+c_2e^{-x}$

                                                              $y(0)=1$

                                                                $y^{'}(0)=2$

                                                                  1. $\hspace{0.2cm}x^2y^{''}-xy^{'}+y=0$

                                                                    $y=c_1x+c_2x\ln \left| x\right|$

                                                                      $y(1)=3$

                                                                        $y^{'}(1)=-1$