Using Charpit’s method, find the complete integral of the differential equation:
p2x + q2y = z
We are asked to solve the first-order partial differential equation (PDE) using Charpit’s method: $$ p^2 x + q^2 y = z, \quad \text{where } p = \frac{\partial z}{\partial x}, \, q = \frac{\partial z}{\partial y}. $$ The Charpit-Lagrange system is given by: $$ \frac{dx}{F_p} = \frac{dy}{F_q} = \frac{dz - p\,dx - q\,dy}{- (p F_p + q F_q)}. $$ Here, $F(x, y, z, p, q) = ______ ________ __________ ________ _____ _______ __________.
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State Euler’s theorem for homogeneous functions and verify it for the function:
Verify that the differential equation:
z(x2-y2-z²)dx+(x+z)xzdy
+ x(z2-x2-xy)dz = 0
is integrable and find its integral.
Verify that the differential equation:
z(x2-y2-z²)dx+(x+z)xzdy
+ x(z2-x2-xy)dz = 0
is integrable and find its integral.
Show that the following function f is differentiable at (0,0):
Find the general solution of the equation:
x(y²-z²)ux + y(z²-x²)uy +z(x²- y²)u₂ = 0
If:
Z= x²y+2xy²
where x = sin zt
and
y = cost
find dz/dt when t=0by (i) chain rule and by (ii) the direct substitution.
Solve the simultaneous equations: