Differential Equations

Q (i) The degree of the differential equation $\frac{{{d^2}y}}{{d{x^2}}} + {e^{dy/dx}} = 0$ is………………. FillBlank Q25 Solve $y + \frac{d}{{dx}}(xy) = x(\sin x + \log x)$. LA Q26 Find the general solution of $(1 + \tan y)(dx - dy) + 2xdy = 0$. LA Q27 Solve $\frac{{dy}}{{dx}} = \cos (x + y) + \sin (x + y)$. LA Q28 Find the general solution of $\frac{{dy}}{{dx}} - 3y = \sin 2x$. LA Q29 Find the equation of a curve passing through (2,1) , if the slope of the tangent to the curve at any point $(x,y)$ is $\frac{{{x^2} + {y^2}}}{{2xy}}$. LA Q31 Find the equation of a curve passing through origin, if the slope of the tangent to the curve at any point $(x,y)$ is equal to the square of the difference of the abcissa and ordinate of the point. LA Q32 Find the equation of a curve passing through the point (1,1) , if the tangent drawn at any point $P(x,y)$
on the curve meets the coordinate axes at $A$ and $B$ such that $P$ is the mid-point of $$AB$$. LA Q33 Solve $x\frac{{dy}}{{dx}} = y(\log y - \log x + 1)$ LA Q34 The degree of the differential equation ${\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2} + {\left( {\frac{{dy}}{{dx}}} \right)^2} = x\sin \left( {\frac{{dy}}{{dx}}} \right)$ is MCQ Q35 The degree of the differential equation ${\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^{3/2}} = \frac{{{d^2}y}}{{d{x^2}}}$ is MCQ Q36 The order and degree of the differential equation$\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^{1/4}} + {x^{1/5}} = 0$
respectively, are MCQ Q37 If $y = {e^{ - x}}(A\cos x + B\sin x)$, then $y$ is a Solution f MCQ Q38 The differential equation for $y = A\cos \alpha x + B\sin \alpha x$, where $A$
and $B$ are arbitrary constants is MCQ Q39 The Solution of differential equation $xdy - ydx = 0$ represents MCQ Q40 The integrating factor of differential equation $\cos x\frac{{dy}}{{dx}} + y\sin x = 1$ is MCQ Q41 The Solution of differential equation $\tan y{\sec ^2}xdx + \tan x{\sec ^2}ydy = 0$ is MCQ Q42 The family $y = Ax + {A^3}$ of curves is represented by differential equation of degree MCQ Q43 The integrating factor of $\frac{{xdy}}{{dx}} - y = {x^4} - 3x$ is MCQ Q44 The Solution of $\frac{{dy}}{{dx}} - y = 1,y(0) = 1$ is given by MCQ Q45 The number of Solution s of $\frac{{dy}}{{dx}} = \frac{{y + 1}}{{x - 1}}$, when $y(1) = 2$ is MCQ Q46 Which of the following is a second order differential equation? MCQ Q47 The integrating factor of differential equation
$\left( {1 - {x^2}} \right)\frac{{dy}}{{dx}} - xy = 1$ is MCQ Q48 ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = C$ is general Solution of the differential equation MCQ Q49 The differential equation $y\frac{{dy}}{{dx}} + x = C$ represents MCQ Q50 The general Solution of ${e^x}\cos ydx - {e^x}\sin ydy = 0$ is MCQ Q51 The degree of differential equation $\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^3} + 6{y^5} = 0$ is MCQ Q52 The Solution of $\frac{{dy}}{{dx}} + y = {e^{ - x}},$ $y(0) = 0$ is MCQ Q53 The integrating factor of differential equation
$\frac{{dy}}{{dx}} + y\tan x - \sec x = 0$ is MCQ Q54 The Solution of differential equation $\frac{{dy}}{{dx}} = \frac{{1 + {y^2}}}{{1 + {x^2}}}$ is MCQ Q55 The integrating factor of differential equation $\frac{{dy}}{{dx}} + y = \frac{{1 + y}}{x}$ is MCQ Q56 $y = a{e^{mx}} + b{e^{ - mx}}$
satisfies which of the following differential equation? MCQ Q56 The Solution of differential equation $\cos x\sin ydx + \sin x\cos ydy = 0$ is MCQ Q58 The Solution of $x\frac{{dy}}{{dx}} + y = {e^x}$ is MCQ Q59 The differential equation of the family of curves ${x^2} + {y^2} - 2ay = 0$ where $a$ is arbitrary constant, is MCQ Q60 The family $Y = Ax + {A^3}$ of curves will correspond to a differential equation of order MCQ Q61 The general Solution of $\frac{{dy}}{{dx}} = 2x{e^{{x^2} - y}}$ is MCQ Q62 The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is MCQ Q63 The general Solution of differential equation $\frac{{dy}}{{dx}} = {e^{\frac{{{x^2}}}{2}}} + xy$ is MCQ Q64 The Solution of equation $(2y - 1)dx - (2x + 3)dy = 0$ is MCQ Q65 The differential equation for which $y = a\cos x + b\sin x$ is a MCQ Q66 The Solution of $\frac{{dy}}{{dx}} + y = {e^{ - x}},y(0) = 0$ is MCQ Q67 The order and degree of differential equation
${\left( {\frac{{{d^3}y}}{{d{x^3}}}} \right)^2} - 3\frac{{{d^2}y}}{{d{x^2}}} + 2{\left( {\frac{{dy}}{{dx}}} \right)^4} = {y^4}$ are MCQ Q68 The order and degree of differential equation

$\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right] = \frac{{{d^2}y}}{{d{x^2}}}$ are MCQ Q69 The differential equation of family of curves
${y^2} = 4a(x + a)$ is MCQ Q70 Which of the following is the general Solution of $\frac{{{d^2}y}}{{d{x^2}}} - 2\frac{{dy}}{{dx}} + y = 0?$ MCQ Q71 The general MCQ Q72 The Solution of differential equation $\frac{{dy}}{{dx}} + \frac{y}{x} = \sin x$ is MCQ Q73 The general Solution of differential equation
$\left( {{e^x} + 1} \right)ydy = (y + 1){e^x}dx$ is MCQ Q74 The Solution of differential equation $\frac{{dy}}{{dx}} = {e^{x - y}} + {x^2}{e^{ - y}}$ is MCQ Q75 The Solution of differential equation

$\frac{{dy}}{{dx}} + \frac{{2xy}}{{1 + {x^2}}} = \frac{1}{{{{\left( {1 + {x^2}} \right)}^2}}}$ is MCQ Q1 Find the solution of $\frac{{dy}}{{dx}} = {2^{y - x}}$. SA Q2 Find the differential equation of all non-vertical lines in a plane. SA Q3 If $\frac{{dy}}{{dx}} = {e^{ - 2y}}$ and $y = 0$ when $x = 5,$ then find the value of $x$ when $y = 3$. SA Q4 Solve $\left( {{x^2} - 1} \right)\frac{{dy}}{{dx}} + 2xy = \frac{1}{{{x^2} - 1}}$. SA Q5 Solve $\frac{{dy}}{{dx}} + 2xy = y$ SA Q6 Find the general solution of $\frac{{dy}}{{dx}} + ay = {e^{mx}}$. SA Q7 Solve the differential equation $\frac{{dy}}{{dx}} + 1 = {e^{x + y}}$. SA Q8 Solve $ydx - xdy = {x^2}ydx$. SA Q9 Solve the differential equation
$\frac{{dy}}{{dx}} = 1 + x + {y^2} + x{y^2},$ when $y = 0$ and $x = 0$. SA Q10 Find the general solution of
$\left( {x + 2{y^3}} \right)\frac{{dy}}{{dx}} = y$. SA Q11 If $y(x)$ is a solution of
$\left( {\frac{{2 + \sin x}}{{1 + y}}} \right)\frac{{dy}}{{dx}} = - \cos x$
and $y(0) = 1,$
then find the value of $y\left( {\frac{\pi }{2}} \right)$. SA Q12 If $y(t)$ is a solution of $(1 + t)\frac{{dy}}{{dt}} - ty = 1$ and
$y(0) = - 1,$ then show that $y(1) = - \frac{1}{2}$. SA Q13 Form the differential equation
having $y = {\left( {{{\sin }^{ - 1}}x} \right)^2} + A{\cos ^{ - 1}}x + B$, where $A$ and $B$
are arbitrary constants, as its general solution. SA Q14 Form the differential equation of all circles which pass through origin and whose centres lie on Y-axis. SA Q15 Find the equation of a curve passing through origin and satisfying the differential equation $\left( {1 + {x^2}} \right)\frac{{dy}}{{dx}} + 2xy = 4{x^2}$. SA Q16 Solve ${x^2}\frac{{dy}}{{dx}} = {x^2} + xy + {y^2}$. SA Q17 Find the general solution of the differential equation

$\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right)\frac{{dy}}{{dx}} = 0$. SA Q18 Find the general solution of ${y^2}dx + \left( {{x^2} - xy + {y^2}} \right)dy = 0$. SA Q19 Solve $(x + y)(dx - dy) = dx + dy$. SA Q20 Solve $2(y + 3) - xy\frac{{dy}}{{dx}} = 0,$ given that $y(1) = - 2$. SA Q21 Solve the differential equation $dy = \cos x(2 - y{\mathop{\rm cosec}\nolimits} x)dx$ given that $y = 2,$ when $x = \frac{\pi }{2}$. SA Q22 Form the differential equation by eliminating $A$ and $B$ in $A{x^2} + B{y^2} = 1$. SA Q23 Solve the differential equation $\left( {1 + {y^2}} \right){\tan ^{ - 1}}xdx + 2y\left( {1 + {x^2}} \right)dy = 0$. SA Q24 Find the differential equation
of system of concentric circles with
centre (1,2). SA

Q1 $\cfrac{{{d^4}y}}{{d{x^4}}} + \sin ({y^m}) = 0$ SA Q2 $y' + 5y = 0$ SA Q3 ${\left( {\cfrac{{ds}}{{dt}}} \right)^4} + 3s\cfrac{{{d^2}s}}{{d{t^2}}} = 0$ SA Q4 ${\left( {\cfrac{{{d^2}y}}{{d{x^2}}}} \right)^2} + \cos \left( {\cfrac{{dy}}{{dx}}} \right) = 0$ SA Q5 $\cfrac{{{d^2}y}}{{d{x^2}}} = \cos 3x + \sin 3x$ SA Q6 ${(y''')^2} + {(y'')^3} + {(y')^4} + {y^5} = 0$ SA Q7 $y''' + 2y'' + y' = 0$ SA Q8 $y' + y = P$ SA Q9 $y'' + {(y')^2} + 2y = 0$ SA Q10 $y'' + 2y' + \sin y = 0$ SA Q11 The degree of the differential equation
${\left( {\cfrac{{{d^2}y}}{{d{x^2}}}} \right)^3} + \left( {\cfrac{{dy}}{{dx}}} \right) + \sin \left( {\cfrac{{dy}}{{dx}}} \right) + 1 = 0$ is

• $3$

• $2$

• $1$

• not defined

SA Q12 The order of the differential equation
$2{x^2}\cfrac{{{d^2}y}}{{d{x^2}}} - 3\cfrac{{dy}}{{dx}} + y = 0$ is

• $2$

• $1$

• $0$

• not defined

SA

Q1 $y = {e^x} + 1:y'' - y' = 0$ SA Q2 $y = {x^2} + 2x + C:y' - 2x - 2 = 0$ SA Q3 $y = \cos x + C:y' + \sin x = 0$ SA Q4 $y = \sqrt {1 + {x^2}} y' = \cfrac{W}{{1 + {x^2}}}$ SA Q5 $y = Ax:xy' = y(x \ne 0)$ SA Q6 $y = x\sin x:xy' = y + x\sqrt {{x^2} - {y^2}} (x \ne 0$ and $x > y$or $x < - y)$ SA Q7 $xy = \log y + C:y' = \cfrac{{{y^2}}}{{1 - xy}}(xy \ne 1)$ SA Q8 $y - \cos y = x:(y\sin y + \cos y + x)y' = y$ SA Q9 $x + y = {\tan ^{ - 1}}y:{y^2}y' + {y^2} + 1 = 0$ SA Q10 $y = \sqrt {{a^2} - {x^2}} ,x \in ( - a,a):x + y\cfrac{{dy}}{{dx}} = 0(y \ne 0)$ SA Q11 The number of arbitrary constants in the general solution of a differential equation of fourth order are:

• $0$

• $2$

• $3$

• $4$

SA Q12 The number of arbitrary constants in the particular solution of a differential equation of third order are:

• $3$

• $2$

• $1$

• $0$

SA

Q1 $\cfrac{x}{a} + \cfrac{y}{b} = 1$ SA Q2 ${y^2} = a({b^2} - {x^2})$ SA Q3 $y = a{e^{3x}} + b{e^{ - 2x}}$ SA Q4 $y = {e^{2x}}\left( {a + bx} \right)$ SA Q5 $y = {e^x}(a\cos x + b\sin x)$ SA Q6 Form the differential equation of the family of circles touching the y-axis at origin SA Q7 Form the differential equation of the family of parabolas having vertex at origin and axis along positive $y$-axis. SA Q8 Form the differential equation of the family of ellipses having foci on $y$-axis and centre at origin. SA Q9 Form the differential equation of the family of hyperbolas having foci on $x$-axis and centre at origin. SA Q10 Form the differential equation of the family of circles having centre on $y$-axis and radius 3 units. SA Q11 Which of the following differential equations has $y = {c_1}{e^x} + {c_2}{e^{ - x}}$ as the general solution?

• $\cfrac{{{d^2}y}}{{d{x^2}}} + y = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} - y = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} + 1 = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} - 1 = 0$

SA Q12 Which of the following differential equations has $y = x$ as one of its particular solution?

• $\cfrac{{{d^2}y}}{{d{x^2}}} - {x^2}\cfrac{{dy}}{{dx}} + xy = x$

• $\cfrac{{{d^2}y}}{{d{x^2}}} + x\cfrac{{dy}}{{dx}} + xy = x$

• $\cfrac{{{d^2}y}}{{d{x^2}}} - {x^2}\cfrac{{dy}}{{dx}} + xy = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} + {x^2}\cfrac{{dy}}{{dx}} + xy = 0$

SA

Q1 $\cfrac{{dy}}{{dx}} = \cfrac{{1 - \cos x}}{{1 + \cos x}}$ SA Q2 $\cfrac{{dy}}{{dx}} = \sqrt {4 - {y^2}} ( - 2 < y < 2)$ SA Q3 $\cfrac{{dy}}{{dx}} + y = 1(y \ne 1)$ SA Q4 ${\sec ^2}x\tan ydx + {\sec ^2}y\tan xdy = 0$ SA Q5 $({e^x} + {e^{ - x}})dy - ({e^x} - {e^{ - x}})dx = 0$ SA Q6 $\cfrac{{dy}}{{dx}} = (1 + {x^2})(1 + {y^2})$ SA Q7 $y\log ydx - xdy = 0$ SA Q8 ${x^5}\cfrac{{dy}}{{dx}} = - {y^5}$ SA Q9 $\cfrac{{dy}}{{dx}} = {\sin ^{ - 1}}x$ SA Q10 ${e^x}\tan ydx + (1 - {e^x}){\sec ^2}ydy = 0$ SA Q11 $({x^3} + {x^2} + x + 1)\cfrac{{dy}}{{dx}} = 2{x^2} + x;y = 1$ when $x = 0$ SA Q12 $x({x^2} - 1)\cfrac{{dy}}{{dx}} = 1;y = 0$ when $x = 2$ SA Q13 $\cos \left( {\cfrac{{dy}}{{dx}}} \right) = a(a \in R);y = 1$ when $x = 0$ SA Q14 $\cfrac{{dy}}{{dx}} = y\tan x;y = 1$ when $x = 0$ SA Q15 Find the equation of a curve passing through the point $(0,0)$ and whose differential equation is$y' = {e^x}\sin x$. SA Q16 For the differential equation $xy\cfrac{{dy}}{{dx}} = (x + 2)(y + 2)$, find the solution curve passing through the point $\left( {1, - 1} \right)$. SA Q17 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. SA Q18 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 $( - 2,1)$. SA Q19 The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of balloon after $t$ seconds. SA Q20 In a bank, principal increases continuously at the rate of $r\%$ per year. Find the value of $r$ if Rs. 100 double itself in 10 years$({\log _e}2 = 0.6931)$. SA Q21 In a bank, principal increases continuously at the rate of $5\%$ per year. An amount of Rs. 1000 is deposited with this bank, how much will it worth after 10 years $({e^{0.5}} = 1.648)$. SA Q22 In a culture, the bacteria count is 1,00,000. The number is increased by 10\% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present? SA Q23 The general solution of the differential equation $\cfrac{{dy}}{{dx}} = {e^{x + y}}$ is

• ${e^x} + {e^{ - y}} = C$

• ${e^x} + {e^y} = C$

• ${e^{ - x}} + {e^y} = C$

• ${e^{ - x}} + {e^{ - y}} = C$

SA

Q1 $({x^2} + xy)dy = ({x^2} + {y^2})dx$ SA Q2 $y' = \cfrac{{x + y}}{x}$ SA Q3 $(x - y)dy - (x + y)dx = 0$ SA Q4 $({x^2} - {y^2})dx + 2xydy = 0$ SA Q5 ${x^2}\cfrac{{dy}}{{dx}} = {x^2} - 2{x^2} + xy$ SA Q6 $xdy - ydx = \sqrt {{x^2} + {y^2}} d\kappa$ SA Q7 $\left\{ {x\cos \left( {\cfrac{y}{x}} \right) + y\sin \left( {\cfrac{y}{x}} \right)} \right\}ydx$ $= \left\{ {y\sin \left( {\cfrac{y}{x}} \right) - x\cos \left( {\cfrac{y}{x}} \right)} \right\}$ SA Q8 $x\cfrac{{dy}}{{dx}} - y + x\sin \left( {\cfrac{y}{x}} \right) = 0$ SA Q9 $ydx + x\log \left( {\cfrac{y}{x}} \right)dy - 2xdy = 0$ SA Q10 $\left( {1 + {e^{x/y}}} \right)dx + {e^{x/y}}\left( {1 - \cfrac{x}{y}} \right)dy = 0$ SA Q11 $\left( {x + y} \right)dy + (x - y)dx = 0;y = 1$ when $x = 1$ SA Q12 ${x^2}dy + (xy + {y^2})dx = 0;y = 1$ when $x = 1$ SA Q13 $\left[ {x{{\sin }^2}\left( {\cfrac{y}{x}} \right) - y} \right]dx + xdy = 0;y = \cfrac{\pi }{4}$ when $x = 1$ SA Q14 $\cfrac{{dy}}{{dx}} - \cfrac{y}{x} + {\rm{cosec}}\left( {\cfrac{y}{x}} \right) = 0;y = 0$ when $x = 1$ SA Q15 $2xy + {y^2} - 2{x^2}\cfrac{{dy}}{{dx}} = 0;y = 2$ when $x = 1$ SA Q16 A homogeneous differential equation of the form $\cfrac{{dx}}{{dy}} = h\left( {\cfrac{x}{y}} \right)$ can be solved by making the substitution

• $y = vx$

• $v = yx$

• $x = vy$

• $x = v$

SA

Q1 $\cfrac{{dy}}{{dx}} + 2y = \sin x$ SA Q2 $\cfrac{{dy}}{{dx}} + 3y = {e^{ - 2x}}$ SA Q3 $\cfrac{{dy}}{{dx}} + \cfrac{y}{x} = {x^2}$ SA Q4 $\cfrac{{dy}}{{dx}} + (\sec x)y = \tan x\left( {0 \le x < \cfrac{\pi }{2}} \right)$ SA Q5 ${\cos ^2}x\cfrac{{dy}}{{dx}} + y = \tan x\left( {0 \le x < \cfrac{\pi }{2}} \right)$ SA Q6 $x\cfrac{{dy}}{{dx}} + 2y = {x^2}\log x$ SA Q7 $x\log x\cfrac{{dy}}{{dx}} + y = \cfrac{2}{x}\log x$ SA Q8 $(1 + {x^2})dy + 2xydx = \cot xdx(x \ne 0)$ SA Q9 $x\cfrac{{dy}}{{dx}} + y - x + xy\cot x = 0(x \ne 0)$ SA Q10 $(x + y)\cfrac{{dy}}{{dx}} = 1$ SA Q11 $ydx + (x - {y^2})dy = 0$ SA Q12 $\left( {x + 3{y^2}} \right)\cfrac{{dy}}{{dx}} = y\left( {y > 0} \right)$ SA Q13 $\cfrac{{dy}}{{dx}} + 2y\tan x = \sin x;y = 0$ when $x = \cfrac{\pi }{3}$ SA Q14 $(1 + {x^2})\cfrac{{dy}}{{dx}} + 2xy = \cfrac{1}{{1 + {x^2}}}$ ; $y = 0$ when $x = 1$ SA Q15 $\cfrac{{dy}}{{dx}} - 3y\cot x = \sin 2x;y = 2$ when $x = \cfrac{\pi }{2}$ SA Q16 Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point $(x,\;y)$ is equal to the sum of the coordinates of the point. SA Q17 Which of the following is a homogeneous differential equation?

• $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$

• $(xy)dx - ({x^3} + {y^3})dy = 0$

• $({x^3} + 2{y^2})dx + 2xydy = 0$

• ${y^2}dx + ({x^2} - xy - {y^2})dy = 0$

SA Q17 Find the equation of a curve passing through the point $(0,2)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5. SA Q18 The Integrating Factor of the differential equation $x\cfrac{{dy}}{{dx}} - y = 2{x^2}$ is

• ${e^{ - x}}$

• ${e^{ - y}}$

• $\cfrac{1}{x}$

• $x$

SA Q19 The Integrating Factor of the differential equation $(1 - {y^2})\cfrac{{dx}}{{dy}} + yx = ay( - 1 < y < 1)$ is

• $\cfrac{1}{{{y^2} - 1}}$

• $\cfrac{1}{{\sqrt {{y^2} - 1} }}$

• $\cfrac{1}{{1 - {y^2}}}$

• $\cfrac{1}{{\sqrt {1 - {y^2}} }}$

SA

Q Find a particular solution of the differential equation
$(x - y)(dx + dy) = dx - dy$,
given that $y = - 1$, when $x = 0$ (Hint: put $x - y = t$) SA Q1 For each of the differential equations given below, indicate its order and degree (if defined).

(i) $\frac{{{d^2}y}}{{d{x^2}}} + 5x{\left( {\frac{{dy}}{{dx}}} \right)^2} - 6y = \log x$

(ii) ${\left( {\frac{{dy}}{{dx}}} \right)^3} - 4{\left( {\frac{{dy}}{{dx}}} \right)^2} + 7y = \sin x$

(iii) $\frac{{{d^4}y}}{{d{x^4}}} - \sin \left( {\frac{{{d^3}y}}{{d{x^3}}}} \right) = 0$ SA Q2 For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i) $y = a{e^x} + b{e^{ - x}} + {x^2}$: $x\frac{{{d^2}y}}{{d{x^2}}} + 2\frac{{dy}}{{dx}} - xy + {x^2} - 2 = 0$

(ii) $y = {e^x}(a\cos x + b\sin x)$: $\frac{{{d^2}y}}{{d{x^2}}} - 2\frac{{dy}}{{dx}} + 2y = 0$

(iii) $y = x\sin 3x$ $:$ $\frac{{{d^2}y}}{{d{x^2}}} + 9y - 6\cos 3x = 0$

(iv) ${x^2} = 2{y^2}\log y$: $\left( {{x^2} + {y^2}} \right)\frac{{dy}}{{dx}} - xy = 0$ SA Q3 Form the differential equation representing the family of curves given by
${(x - a)^2} + 2{y^2} = {a^2}$
where ${\rm{a}}$ is an arbitrary constant. SA Q4 Prove that ${x^2} - {y^2} = c{\left( {{x^2} + {y^2}} \right)^2}$ is the general solution of differential equation, $\left( {{x^3} - 3x{y^2}} \right)dx = \left( {{y^3} - 3{x^2}y} \right)dy$ where $c$ is a parameter. SA Q5 Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes. SA Q6 Find the general solution of the differential equation $\frac{{dy}}{{dx}} + \sqrt {\frac{{1 - {y^2}}}{{1 - {x^2}}}} = 0$ SA Q7 Show that the general solution of the differential equation $\frac{{dy}}{{dx}} + \frac{{{y^2} + y + 1}}{{{x^2} + x + 1}} = 0$ is given by $(x + y + 1) = A(1 - x - y - 2xy)$, where $A$ is parameter SA Q8 Find the equation of the curve passing through the point $\left( {0,\frac{\pi }{4}} \right)$ whose differential equation is, $\sin x\cos ydx + \cos x\sin ydy = 0$ SA Q9 Find the particular solution of the differential equation
$\left( {1 + {e^{2x}}} \right)dy + \left( {1 + {y^2}} \right){e^x}dx = 0$,
given that $y = 1$ when $x = 0$ SA Q10 Solve the differential equation $y{e^{\frac{x}{y}}}dx = \left( {x{e^{\frac{x}{y}}} + {y^2}} \right)dy(y \ne 0)$ SA Q12 Solve the differential equation $\left[ {\frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \frac{y}{{\sqrt x }}} \right]\frac{{dx}}{{dy}} = 1(x \ne 0)$ SA Q13 Find a particular solution of the differential equation $\frac{{dy}}{{dx}} + y\cot x = 4x{\mathop{\rm cosec}\nolimits} x(x \ne 0)$,
given that $y = 0$ when $x = \frac{\pi }{2}$ SA Q14 Find a particular solution of the differential equation
$(x + 1)\frac{{dy}}{{dx}} = 2{e^{ - y}} - 1$
given that $y = 0$ when $x = 0$ SA Q15 The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009 ? SA Q16 The general solution of the differential equation $\frac{{ydx - xdy}}{y} = 0$ is

A. $xy = C$

B. $x = C{y^2}$

C. $y = Cx$

D. $y = C{x^2}$ SA Q17 The general solution of a differential equation of the type $\frac{{dx}}{{dy}} + {{\rm{P}}_1}x = {{\rm{Q}}_1}$, is

A. $y{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dy}}} \right)} dy + {\rm{C}}$

B. $y \cdot {e^{\int {{\rm{P}}_1^{}dx} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{{\rm{P}}_1}} dx}}} \right)} dx + {\rm{C}}$

C. $x{e^{\int {{\rm{P}}_1^{}dy} }} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dy}}} \right)} dy + {\rm{C}}$

D. $x{e^{\int {{\rm{P}}_1^{}} dx}} = \int {\left( {{{\rm{Q}}_1}{e^{\int {{\rm{P}}_1^{}} dx}}} \right)} dx + {\rm{C}}$ SA Q18 The general solution of the differential equation ${e^x}dy + \left( {y{e^x} + 2x} \right)dx = 0$is

A. $x{e^y} + {x^2} = C$

B. $x{e^y} + {y^2} = C$

C. $y{e^x} + {x^2} = c$

D. $y{e^y} + {x^2} = C$ SA