a) Obtain a complete integral of the equation zpq=p+q 6mks b) Verify that the differential equation (y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0 is integrable. 7 mks Question Four: 13 Marks a) Use Charpits method to find the complete integral of the partial differential equation z=px+qy+3(pq)^1/3 b) Solve the equation x^2y^2z^2dx+b^2x^2z^2dy+c^2x^2y^2dz=0 7 mks 6mks Question Five: 13 Marks a) Solve the differential equation (x^2z-y^3)dx+3xy^2dy+x^3dz=0 6mks b) Verify that the equation x(y^2-a^2)dx+y(x^2-z^2)dy-z(y^2-a^2)dz=0 is integrable and hence solve it. 7 mks Question Six: 13 Marks a) Given the equation f(x,y,z,p,q)=0 Define the aspect of compatibility. b) State a necessary and sufficient condition for the Pfaffian equation bar (x)cdot dbar (x)=0 to be integrable 2mks c) Show that the equation xp-yq=0 1mk z(xp+yq)=2xy are compatible and obtain the solution 10mks Question Seven: 13 Marks a) Define a homogenous equation 2mks b) Obtain the complete integral of the partial differential equation p^2x+q^2y=z by Jacob's method 11 mks

Question One:<br />a) To obtain a complete integral of the equation $zpq=p+q$, we can use the method of characteristics. First, we find the characteristic equations by differentiating the given equation with respect to $p$ and $q$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the complete integral.<br /><br />b) To verify that the differential equation $(y^{2}+yz)dx+(xz+z^{2})dy+(y^{2}-xy)dz=0$ is integrable, we need to check if it is an exact differential equation. We can do this by finding the partial derivatives of the coefficients with respect to the other variables and checking if they satisfy the condition for exactness. If the equation is exact, it is integrable.<br /><br />Question Two:<br />a) To find the complete integral of the partial differential equation $z=px+qy+3(pq)^{1/3}$ using Charpits method, we first find the characteristic equations by differentiating the given equation with respect to $p$ and $q$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the complete integral.<br /><br />b) To solve the equation $x^{2}y^{2}z^{2}dx+b^{2}x^{2}z^{2}dy+c^{2}x^{2}y^{2}dz=0$, we can use the method of characteristics. First, we find the characteristic equations by differentiating the given equation with respect to $x$, $y$, and $z$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the solution.<br /><br />Question Three:<br />a) To solve the differential equation $(x^{2}z-y^{3})dx+3xy^{2}dy+x^{3}dz=0$, we can use the method of characteristics. First, we find the characteristic equations by differentiating the given equation with respect to $x$, $y$, and $z$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the solution.<br /><br />b) To verify that the equation $x(y^{2}-a^{2})dx+y(x^{2}-z^{2})dy-z(y^{2}-a^{2})dz=0$ is integrable and hence solve it, we can use the method of characteristics. First, we find the characteristic equations by differentiating the given equation with respect to $x$, $y$, and $z$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the solution.<br /><br />Question Four:<br />a) To define the aspect of compatibility for the equation $f(x,y,z,p,q)=0$, we need to check if there exists a function $\phi(x,y,z,p,q)$ such that $\nabla\phi$ is a solution to the given equation. If such a function exists, then the equation is compatible.<br /><br />b) A necessary and sufficient condition for the Pfaffian equation $\bar{x}\cdot d\bar{x}=0$ to be integrable is that the coefficients of $dx$, $dy$, and $dz$ are exact differentials.<br /><br />c) To show that the equation $xp-yq=0$ and $z(xp+yq)=2xy$ are compatible, we can use the method of characteristics. First, we find the characteristic equations by differentiating the given equations with respect to $x$, $y$, and $z$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the solution.<br /><br />Question Five:<br />a) A homogeneous equation is an equation in which all terms are of the same degree.<br /><br />b) To obtain the complete integral of the partial differential equation $p^{2}x+q^{2}y=z$ by Jacob's method, we first find the characteristic equations by differentiating the given equation with respect to $p$ and $q$ and setting them equal to zero. Solving these equations, we get the characteristic curves. Then, we integrate along these curves to obtain the complete integral.