Differential Equations Test

Differential Equations

This is Differential Equations Test-04 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

Differential equation of the family of circles touching the y-axis at origin is

A
\(2y\prime {\text{ }} - {\text{ }}{x^2} = {\text{ }}{y^2}\)
B
\(2xyy\prime {\text{ }} - {\text{ }}{x^2} = {\text{ }}{y^2}\)
C
\(2xyy\prime {\text{ }} + {\text{ }}{x^2} = {\text{ }}{y^2}\)
Correct
D
\(\begin{array}{*{20}{l}} {2yy\prime {\text{ }} + {\text{ }}{x^2} = {\text{ }}{y^2}} \end{array}\)

Differential equation of the family of parabolas having vertex at origin and axis along positive y-axis is

A
xy′ + 2y = 0
B
\(\begin{array}{*{20}{l}} {xy{\prime ^2} + {\text{ }}2y{\text{ }} = {\text{ }}0} \end{array}\)
C
y′ – 2y = 0
D
xy′ – 2y = 0
Correct

Differential equation of the family of ellipses having foci on y-axis and centre at origin is

A
\(xyy\prime\prime \text{ }-x\left( y\prime \right){}^\text{2}\text{ }+yy\prime \text{ }=\text{ }0\)
B
\(yy\prime\prime \text{ }+\text{ }x\left( y\prime \right){}^\text{2}\text{ }\text{ }yy\prime \text{ }=\text{ }0\)
C
\(xyy\prime\prime \text{ }+\text{ }x\left( y\prime \right){}^\text{2}\text{ }\text{ }yy\prime \text{ }=\text{ }0\)
Correct
D
\(xy\prime\prime \text{ }+\text{ }x\left( y\prime \right){}^\text{2}\text{ }\text{ }yy\prime \text{ }=\text{ }0 \)

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = ex sin x.

A
\(\begin{array}{*{20}{l}} {2y{\text{ }} + {\text{ }}1{\text{ }} = {\text{ }}{e^x}\left( {{\text{ }}sin2{\text{ }}x{\text{ }}--{\text{ }}cos{\text{ }}x} \right)} \end{array}\)
B
\(2y{\text{ }}--{\text{ }}1{\text{ }} = {\text{ }}{e^x}\left( {{\text{ }}sin{\text{ }}x{\text{ }}--{\text{ }}cos{\text{ }}x} \right)\)
Correct
C
\(\begin{array}{*{20}{l}} {3y{\text{ }}--{\text{ }}1{\text{ }} = {\text{ }}{e^x}\left( {{\text{ }}sin{\text{ }}x{\text{ }}--{\text{ }}cos2x} \right)} \end{array}\)
D
\(4y{\text{ }}--{\text{ }}1{\text{ }} = {\text{ }}{e^x}\left( {{\text{ }}sin{\text{ }}x{\text{ }}--{\text{ }}cos{\text{ }}2x} \right)\)

Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

A
\({y^2}--{\text{ }}{x^3} = {\text{ }}4\)
B
\({y^2}--{\text{ }}{x^2} = {\text{ }}4\)
Correct
C
\({y^3}--{\text{ }}{x^2} = {\text{ }}4\)
D
\({y^3}--{\text{ }}{x^3} = {\text{ }}4\)

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

A
\({\left( {x{\text{ }} + {\text{ }}3} \right)^2} = {\text{ }}y{\text{ }} + {\text{ }}4\)
B
\({\left( {x{\text{ }} + {\text{ }}5} \right)^2} = {\text{ }}2y{\text{ }} + {\text{ }}3\)
C
\(\begin{array}{*{20}{l}} {{{\left( {x{\text{ }} + {\text{ }}5} \right)}^2} = {\text{ }}2y{\text{ }} + {\text{ }}3} \end{array}\)
D
\(\begin{array}{*{20}{l}} {{{\left( {x{\text{ }} + {\text{ }}4} \right)}^2} = {\text{ }}y{\text{ }} + {\text{ }}3} \end{array}\)
Correct

Solution of \(x\;dy - ydx = \;\sqrt {{x^2} + {y^2}\;} dx\) is

A
\(y + \sqrt {{x^2} + {y^2}} = C{x^2}\)
Correct
B
\(y + \sqrt {{x^3} + {y^2}} = C{x^2}\)
C
\(y + \sqrt {3 + {y^2}} = C{x^4}\)
D
\(y + \sqrt {{x^2} + {y^3}} = C{x^2}\)

Variable separation method can be used to solveFirst Order, First Degree Differential Equations in which y’ is of the form.

A
y’ = h(x)g(y)
Correct
B
\(y{{}^{2}}=\text{ }sin\text{ }\left( h\left( x \right) \right)\)
C
\(y{{}^{3}}=\text{ }g\left( y \right)\)
D
\(y{{}^{2}}=\text{ }cos\text{ }\left( g\left( y \right) \right)\)

Solution of\(\left\{ {x\cos \left( {\frac{y}{x}} \right) + y\sin \left( {\frac{y}{x}} \right)} \right\}ydx = \left\{ {y\sin \left( {\frac{y}{x}} \right) - \;x\cos \left( {\frac{y}{x}} \right)} \right\}xdy\) is

A
\(x\;\cos \left| {\frac{y}{x}} \right| = C\)
B
\(xy\;\cos \left| {\frac{{3y}}{x}} \right| = C\)
C
\(xy\;\cos \left| {\frac{y}{x}} \right| = C\)
Correct
D
\(y\;\cos \left| {\frac{y}{x}} \right| = C\)

A differential equation of the form y' = F(x,y) is homogeneous if

A
F(x,y) is a homogeneous function of degree two
B
F(x,y) is a homogeneous function of degree zero
Correct
C
F(x,y) is a homogeneous function of degree one
D
F(x,y) is a homogeneous function of degree three

Solution of \(\left( {{x^2}--{\text{ }}{y^2}} \right){\text{ }}dx{\text{ }} + {\text{ }}2xy{\text{ }}dy{\text{ }} = {\text{ }}0\) is

A
\({x^2} + {y^2} = Cx\)
Correct
B
\({x^2} - {y^2} = Cx\)
C
\({x^3} + {y^2} = Cx\)
D
\({x^2} + {y^3} = Cx\)

Find the particular solution for \({x^2}dy + (xy + {y^2})dx = 0 \); y=1 when x =1

A
\(y - 2x = 3{x^{23}}y\)
B
\(y - 2x = 3{x^2}y\)
C
\(y + 2x = - 3{x^2}y\)
D
\(y + 2x = 3{x^2}y\)
Correct

Find the particular solution for\(\frac{{dy}}{{dx}} - \;\frac{y}{x} + {\text{cosec}}\left( {\frac{y}{x}} \right) = 0;y = 0\;when\;x = 1\)

A
\(\cos \left( {\frac{y}{{2x}}} \right) = \;\log \left| {3ex} \right|\)
B
\(\cos \left( {\frac{y}{x}} \right) = \;\log \left| {ex} \right|\)
Correct
C
\(\cos \left( {\frac{{2y}}{x}} \right) = \;\log \left| {ex} \right|\)
D
\(\cos \left( {\frac{y}{x}} \right) = \;\log \left| {2ex} \right|\)

Find the particular solution for \(2xy + {y^2} - 2{x^2}\frac{{dy}}{{dx}}\; = 0;\;y = 2\;when\;x = 1\)

A
\(y = \frac{{2x}}{{1 - \log \left| x \right|}}\;\left( {x\; \ne 0,x \ne e} \right)\)
Correct
B
\(y = \frac{{5x}}{{1 + \log \left| x \right|}}\;\left( {x\; \ne 0,x \ne e} \right)\)
C
\(y = \frac{{2x}}{{1 + \log \left| x \right|}}\;\left( {x\; \ne 0,x \ne e} \right)\)
D
\(y = \frac{{3x}}{{1 - \log \left| x \right|}}\;\left( {x\; \ne 0,x \ne e} \right)\)

A homogeneous equation of the form \(\frac{{dy}}{{dx}} = h\left( {\frac{x}{y}} \right)\) can be solved by making the substitution

A
\(x = \;\nu y\)
Correct
B
\(y = \;\nu x\)
C
\(x = \;\nu \)
D
\(\nu = \;yx\)