Question

 Show that the differential equation:

(y^{2}+yx)dx+(z^{2}+zx)dy+(y^{2}-xy)dz=0

is integrable and find its integral.

03 Mar 2024
Answer :
Word Count : 400
To solve the given differential equation: \[ (y^2 + yx)dx + (z^2 + zx)dy + (y^2 - xy)dz = 0 \] we need to check its integrability and find the integral. ### Step 1: Check for exactness A differential equation is exact if it can be written as \( M(x, y, z)dx + N(x, y, z)dy + P(x, y, z)dz = 0 \), where: \[ \frac{\partial _______ ___ ______ __________ _________ ___ _____ __________ _____ _____ _____.
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 Show that the differential equation:

(y^{2}+yx)dx+(z^{2}+zx)dy+(y^{2}-xy)dz=0

is integrable and find its integral.

1. State whether the following statements are true or false. Give a short proof or a counterexample in support of your answer:                                                  

a) A real-valued function of three variables which is continuous everywhere is differentiable. 

b) The function f\left ( x \right )=ln\left ( \frac{x+y}{x} \right ) is not homogeneous function.

c) The cylindrical coordinates of the point whose spherical coordinates is \left ( 8,\frac{\pi }{6},\frac{\pi }{2} \right )\: is\: \! \! \left ( 8,\frac{\pi }{6},0 \right ).

d) The unique solution y(x) of an ordinary differential equation:

                \frac{dy}{dx}=\frac{dy}{dx}= \left\{\begin{matrix} 0, for \,\, \, x< 0 & \\ 1, for\, \, x\geq 0 & \end{matrix}\right.

    exists \forall x\in \mathbb{R}.

e) The differential equation   \left [ 1+\left ( y \right )^{2} \right ]^{\frac{5}{3}}=y  is a second order differential equation of degree 3.

f) Differential equation:cos\: x\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+xy^{2}=0

           in ]0,\pi[  is a linear, homogeneous equation.

g) The total differential equation corresponding to the family of surfaces , x^{3}z+x^{2}y=c, where c is a parameter is .3x^{2}dz+x\left ( ydx+xdy \right )=0.

h) Differential equation:

     5x^{2}y^{2}z^{2}=2px^{2}y^{2}+5qx^{2}y^{3}+2pz^{2}+9x^{2}y^{2}

     is a semi- linear partial differential equation of  first order.

i) The differential equation:

                        v\frac{du}{dy}=e^{2v}+uv-u     

has an integrating factor v exp(−v) .

j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is:\frac{dx}{y}=\frac{dy}{-w^{2}x}=y_{0}

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