MTH401 Quiz 1 Set 2

Q1. The differential equation $$\begin{array}{l} y(1 + 2xy)dx + x(1 - 2xy)dy = 0\\ \end{array}$$ is

1. None of these
2. Separable
3. Exact
4. Homogeneous

Correct answer: None of these

Q2. $${\text{ydx - y(sinx)dy = 0, is an example of - - - differential equation}}{\text{.}}$$

1. Non-exact
2. Non-homogeneous
3. Exact
4. Non-linear

Correct answer: Non-exact

Q3. Classify the following differential equation $${e^x}\frac{{dy}}{{dx}} + 2y = 3xy$$

1. Linear and not separable
2. Separable and not linear
3. Neither separable nor linear
4. Both separable and linear

Correct answer: Both separable and linear

Q4. $$Which ~of ~the ~following ~substitution~ will~ transform~ the~ differential~ equation:\frac{{dy}}{{dx}} = \frac{y}{x} + \sec \left( {\frac{y}{x}} \right),~in ~to ~separable~ form?$$

1. $$x = vy$$
2. $$y = vx$$
3. $$y = v + x$$
4. $$y = v - x$$

Correct answer: $$y = vx$$

Q5. $$For~ f(x,y) = \frac{{2x}}{{3y}} + 7,f(tx,ty) = - - -$$

1. $$f(x,y)$$
2. $${t^3}f(x,y)$$
3. $$tf(x,y)$$
4. $tf(x,y)$

Correct answer: $$f(x,y)$$

Q6. $$Integrating~factor~for~the~first~order~linear~differential~equation~:{\left( {x - 1} \right)^3}\frac{{dy}}{{dx}} + 4{\left( {x - 1} \right)^2}y = \cos~ x~is - - -$$

1. $${e^{x - 1}}$$
2. $${e^{{1}{{x - 1}}}}$$
3. $$ (x - 1)$$
4. $${(x - 1)^4}$$

Correct answer: $${(x - 1)^4}$$

Q7. The differential equation $$\frac{{dy}}{{dx}} - y = {y^3}$$ may be called as linear equation of

1. Bessel
2. Cauchy
3. Homogeneous
4. Bernoulli’s

Correct answer: Bernoulli’s

Q8. $$\begin{gathered} {\text{A differential equation M(x,y)dx + N(x,y)dy = 0 is exact if there exists a}} \hfill \\\ {\text{multi - variable function f(x,y) such that - - - }} \hfill \\\\ \end{gathered}$$

1. $$df( {x,y} ) = {{ f}}{{ x}}dx + {{ f}}{{ y}}dy$$
2. $$f(x,y) = {{{ f}}{{ x}}} dx + {{{ f}}{{ y}}} dy$$
3. $$f(x,y) = {{ f}}{{ x}}dx + {{ f}}{{ y}}dy$$
4. $$ f (x,y)dx = {{{ f}}{{ x}}} dx + {{{ f}}{{ y}}} dy$$

Correct answer: $$df( {x,y} ) = {{ f}}{{ x}}dx + {{ f}}{{ y}}dy$$

Q9. The general linear ordinary differential equation of order $n$ is: $a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots +a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0$,$\,$where -------.

1. $a_{n}(x)=0$
2. $a_{n}(x) 0$
3. $a_{0}(x)=0$
4. $a_{0}(x) 0$

Correct answer: $a_{n}(x) 0$

Q10. For the nonexact differential equation $$y\left( {xy + 2{x^2}{y^2}} \right)dx + x\left( {xy - {x^2}{y^2}} \right)dy = 0,ifxM(x,y) - yN(x,y) = {x^3}{y^3} \ne 0,$$then the integrating factor is - - -

1. $${-1}{{{x^3}{y^3}}}$$
2. $${1}{{{x^3}{y^3}}}$$
3. $${e^{ - {x^3}{y^3}}}$$
4. $${e^{{x^3}{y^3}}}$$

Correct answer: $${1}{{{x^3}{y^3}}}$$