Engineering Mat...

The order and degree of the differential equation \({\left( {\frac{{{d^4}y}}{{d{x^4}}}} \right)^{\frac{1}{2}}} = {\left[ {1 + {{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)}^2}} \right]^{\frac{1}{3}}}\) respectively are

The particular integral of \({x^2}\frac{{{d^2}y}}{{d{x^2}}} - x\frac{{dy}}{{dx}} + y = 4\log x\) is

The partial differential equation \(\frac{{\partial u}}{{\partial t}} = \alpha \frac{{{\partial ^2}u}}{{\partial {x^2}}}\) where, α is a positive constant is

The solution of the equation \({\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + {\rm{y}} = 0{\rm{}}\) passing through the point (1,1) is

The solution of differential equation

If an integral curve of the differential equation \(\left( {y - x} \right)\frac{{dy}}{{dx}} = 1\) pass through (0, 0) and (α, 1), then α is equal to

Solve \(\frac{{{d^2}y}}{{d{y^2}}} - 3\frac{{dy}}{{dx}} + 2y = x{e^{3x}}\)

If x^r is on integrating factor of (x + y³) dx + 6xy² dy = 0, then r is ______

Given the ordinary differential equation \(\frac{{{d^2}y}}{{d{x^2}}} + \frac{{dy}}{{dx}} - 6y = 0\) With y (0) = 0 and \(\frac{{dy}}{{dx}}\left( 0 \right) = 1\), the value of y(1) is ______ (correct to two decimal places).

The differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants is: