How to solve this PDE using Charpit's Method?

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If $p=fracpartial zpartial x$ and $q=fracpartial zpartial y$ then consider the following PDE:
$$p^2+q=0.$$



The using Charpit's Method we get the following auxiliary equations:
$$fracdxdt =2p$$
$$ fracdydt = 1 $$
$$ fracdzdt = 2p^2+q = p^2$$
$$ fracdpdt = 0$$
$$fracdqdt = 0.$$



Thus $p= c_1$ and $q= c_2$ with $c_2^2+c_1 = 0.$ Then
$$ x= 2c_1t+a_1 $$
$$y = t+a_2$$
$$z = tc_1^2+a_3.$$



Now I have the following initial conditions $z(x,0)=xd.$ How do I get the final solution? In particular, I am not sure how to eliminate $t$ from the three equations.










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    If $p=fracpartial zpartial x$ and $q=fracpartial zpartial y$ then consider the following PDE:
    $$p^2+q=0.$$



    The using Charpit's Method we get the following auxiliary equations:
    $$fracdxdt =2p$$
    $$ fracdydt = 1 $$
    $$ fracdzdt = 2p^2+q = p^2$$
    $$ fracdpdt = 0$$
    $$fracdqdt = 0.$$



    Thus $p= c_1$ and $q= c_2$ with $c_2^2+c_1 = 0.$ Then
    $$ x= 2c_1t+a_1 $$
    $$y = t+a_2$$
    $$z = tc_1^2+a_3.$$



    Now I have the following initial conditions $z(x,0)=xd.$ How do I get the final solution? In particular, I am not sure how to eliminate $t$ from the three equations.










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      If $p=fracpartial zpartial x$ and $q=fracpartial zpartial y$ then consider the following PDE:
      $$p^2+q=0.$$



      The using Charpit's Method we get the following auxiliary equations:
      $$fracdxdt =2p$$
      $$ fracdydt = 1 $$
      $$ fracdzdt = 2p^2+q = p^2$$
      $$ fracdpdt = 0$$
      $$fracdqdt = 0.$$



      Thus $p= c_1$ and $q= c_2$ with $c_2^2+c_1 = 0.$ Then
      $$ x= 2c_1t+a_1 $$
      $$y = t+a_2$$
      $$z = tc_1^2+a_3.$$



      Now I have the following initial conditions $z(x,0)=xd.$ How do I get the final solution? In particular, I am not sure how to eliminate $t$ from the three equations.










      share|cite|improve this question













      If $p=fracpartial zpartial x$ and $q=fracpartial zpartial y$ then consider the following PDE:
      $$p^2+q=0.$$



      The using Charpit's Method we get the following auxiliary equations:
      $$fracdxdt =2p$$
      $$ fracdydt = 1 $$
      $$ fracdzdt = 2p^2+q = p^2$$
      $$ fracdpdt = 0$$
      $$fracdqdt = 0.$$



      Thus $p= c_1$ and $q= c_2$ with $c_2^2+c_1 = 0.$ Then
      $$ x= 2c_1t+a_1 $$
      $$y = t+a_2$$
      $$z = tc_1^2+a_3.$$



      Now I have the following initial conditions $z(x,0)=xd.$ How do I get the final solution? In particular, I am not sure how to eliminate $t$ from the three equations.







      linear-pde






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      asked Aug 30 at 3:49









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