1. The general solution of the differential equation, $$\frac{{{{\text{d}}^4}{\text{y}}}}{{{\text{d}}{{\text{x}}^4}}} - 2\frac{{{{\text{d}}^3}{\text{y}}}}{{{\text{d}}{{\text{x}}^3}}} + 2\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = 0$$ is
2. The solution of the differential equation, $${\text{y}}\sqrt {1 - {{\text{x}}^2}} {\text{dy}} + {\text{x}}\sqrt {1 - {{\text{y}}^2}} {\text{dx}} = {\text{0}}$$ is
3. The maximum value of the solution y(t) of the differential equation \[{\rm{y}}\left( {\rm{t}} \right) + {\rm{\ddot y}}\left( {\rm{t}} \right) = 0\] with initial conditions \[{\rm{\dot y}}\left( 0 \right) = 1\] and y(0) = 1, for t ≥ 0 is
4. The order of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{t}}^2}}} + {\left( {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right)^3} + {{\text{y}}^4} = {{\text{e}}^{ - {\text{t}}}}$$ is
5. The order and degree of the differential equation $$\frac{{{{\text{d}}^3}{\text{y}}}}{{{\text{d}}{{\text{x}}^3}}} + 4\sqrt {{{\left( {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right)}^3} + {{\text{y}}^2}} = 0$$ are respectively
6. Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t2
The particular solution of the differential equation is
The particular solution of the differential equation is
7. The differential equation $${\left[ {1 + {{\left( {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right)}^2}} \right]^3} = {{\text{C}}^2}{\left[ {\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}}} \right]^2}$$ is of
8. Consider a system governed by the following equations:
$$\frac{{{\text{d}}{{\text{x}}_1}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_2}\left( {\text{t}} \right) - {{\text{x}}_1}\left( {\text{t}} \right);\,\frac{{{\text{d}}{{\text{x}}_2}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_1}\left( {\text{t}} \right) - {{\text{x}}_2}\left( {\text{t}} \right)$$
The initial conditions are such that $${{\text{x}}_1}\left( 0 \right) < {{\text{x}}_2}\left( 0 \right) < \infty .$$ Let $${{\text{x}}_{1{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_1}\left( {\text{t}} \right)$$ and $${{\text{x}}_{2{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_2}\left( {\text{t}} \right).$$ Which one of the following is true?
$$\frac{{{\text{d}}{{\text{x}}_1}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_2}\left( {\text{t}} \right) - {{\text{x}}_1}\left( {\text{t}} \right);\,\frac{{{\text{d}}{{\text{x}}_2}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_1}\left( {\text{t}} \right) - {{\text{x}}_2}\left( {\text{t}} \right)$$
The initial conditions are such that $${{\text{x}}_1}\left( 0 \right) < {{\text{x}}_2}\left( 0 \right) < \infty .$$ Let $${{\text{x}}_{1{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_1}\left( {\text{t}} \right)$$ and $${{\text{x}}_{2{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_2}\left( {\text{t}} \right).$$ Which one of the following is true?