Q77(v):

State True or False: Number of arbitrary constants in the particular solution of a differential equation of order two is two.

More Questions from Class 12 Mathematics Differential Equations EXERCISE 9.3

• Q1: Find the solution of $\frac{dy}{dx} = \frac{2y}{x}$.
• Q2: Find the differential equation of all non vertical lines in a plane.
• Q3: Given that $\frac{dy}{dx} = e^{-2y}$ and $y = 0$ when $x = 5$. Find the value of $x$ when $y = 3$.
• Q4: Solve the differential equation $(x^2 – 1) \frac{dy}{dx} + 2xy = \frac{1}{x^2 - 1}$.
• Q5: Solve the differential equation $\frac{dy}{dx} + 2xy = y^2$.
• Q6: Find the general solution of $\frac{dy}{dx} + ay = e^{mx}$.
• Q7: Solve the differential equation $\frac{dy}{dx} + 1 = e^{x+y}$.
• Q8: Solve: $ydx – xdy = x^2ydx$.
• Q9: Solve the differential equation $\frac{dy}{dx} = 1 + x + y^2 + xy^2$, when $y = 0$, $x = 0$.
• Q10: Find the general solution of $(x + 2y^3)\frac{dy}{dx} = y$.
• Q11: If $y(x)$ is a solution of $\frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{1}{(1+x^2)^2}$ and $y (0) = 1$, then find the value of $y(\frac{\pi}{2})$.
• Q12: If $y(t)$ is a solution of $(1 + t)\frac{dy}{dt} – ty = 1$ and $y (0) = – 1$, then show that $y (1) = –\frac{1}{2}$.
• Q13: Form the differential equation having $y = (\sin^{-1}x)^2 + A\cos^{-1}x + B$, where $A$ and $B$ are arbitrary constants, as its general solution.
• Q14: Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
• Q15: Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx} + 2xy = 4x^2$.
• Q16: Solve : $x^2\frac{dy}{dx} = x^2 + xy + y^2$.
• Q17: Find the general solution of the differential equation $(1 + y^2) + (x – e^{\tan^{-1}y})\frac{dy}{dx} = 0$.
• Q18: Find the general solution of $y^2dx + (x^2 – xy + y^2) dy = 0$.
• Q19: Solve : $(x + y) (dx – dy) = dx + dy$.[Hint: Substitute $x + y = z$ after seperating $dx$ and $dy$]
• Q20: Solve : $2 (y + 3) – xy\frac{dy}{dx} = 0$, given that $y (1) = – 2$.
• Q21: Solve the differential equation $dy = \cos x (2 – y \csc x) dx$ given that $y = 2$ when $x = \frac{\pi}{2}$.
• Q22: Form the differential equation by eliminating $A$ and $B$ in $Ax^2 + By^2 = 1$.
• Q23: Solve the differential equation $(1 + y^2) \tan^{-1}x dx + 2y (1 + x^2) dy = 0$.
• Q24: Find the differential equation of system of concentric circles with centre (1, 2).
• Q25: Solve : $\frac{d}{dx}(y + xy) = x (\sin x + \log x)$
• Q26: Find the general solution of $(1 + \tan y) (dx – dy) + 2xdy = 0$.
• Q27: Solve : $\frac{dy}{dx} = \cos(x + y) + \sin (x + y)$.[Hint: Substitute $x + y = z$]
• Q28: Find the general solution of $\frac{dy}{dx} - 3y = \sin 2x$.
• 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 $\frac{x^2 + y^2}{2xy}$.
• Q30: Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point $(x, y)$ is $\frac{y-1}{x^2+x}$.
• 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.
• 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 co-ordinate axes at A and B such that P is the mid-point of AB.
• Q33: Solve : $x\frac{dy}{dx} = y (\log y – \log x + 1)$
• Q34: The degree of the differential equation $(\frac{d^2y}{dx^2})^2 + (\frac{dy}{dx})^2 = x \sin(\frac{dy}{dx})$ is:
• Q35: The degree of the differential equation $[1 + (\frac{dy}{dx})^2]^{\frac{3}{2}} = \frac{d^2y}{dx^2}$ is
• Q36: The order and degree of the differential equation $(\frac{d^2y}{dx^2})^{\frac{1}{2}} + (\frac{dy}{dx})^{\frac{1}{5}} + 4 = 0$, respectively, are
• Q37: If $y = e^{-x} (A\cos x + B\sin x)$, then $y$ is a solution of
• Q38: The differential equation for $y = A\cos \alpha x + B\sin \alpha x$, where $A$ and $B$ are arbitrary constants is
• Q39: Solution of differential equation $xdy – ydx = 0$ represents :
• Q40: Integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is :
• Q41: Solution of the differential equation $\tan y \sec^2x dx + \tan x \sec^2y dy = 0$ is :
• Q42: Family $y = Ax + A^3$ of curves is represented by the differential equation of degree :
• Q43: Integrating factor of $x\frac{dy}{dx} – y = x^4 – 3x$ is :
• Q44: Solution of $\frac{dy}{dx} - y = 1$, $y (0) = 1$ is given by
• Q45: The number of solutions of $\frac{dy}{dx} = \frac{y+1}{x-1}$ when $y (1) = 2$ is :
• Q46: Which of the following is a second order differential equation?
• Q47: Integrating factor of the differential equation $(1 – x^2)\frac{dy}{dx} - xy = 1$ is
• Q48: $\tan^{-1} x + \tan^{-1} y = c$ is the general solution of the differential equation:
• Q49: The differential equation $y\frac{dy}{dx} + x = c$ represents :
• Q50: The general solution of $e^x \cos y dx – e^x \sin y dy = 0$ is :
• Q51: The degree of the differential equation $(\frac{d^2y}{dx^2})^2 + (\frac{dy}{dx})^3 + y^5 + 6 = 0$ is :
• Q52: The solution of $\frac{dy}{dx} + y = e^{-x}, y(0) = 0$ is :
• Q53: Integrating factor of the differential equation $\frac{dy}{dx} + y \tan x - \sec x = 0$ is:
• Q54: The solution of the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ is:
• Q55: The integrating factor of the differential equation $\frac{dy}{dx} + (1+\frac{1}{x})y = ...$ is:
• Q56: $y = ae^{mx} + be^{-mx}$ satisfies which of the following differential equation?
• Q57: The solution of the differential equation $\cos x \sin y dx + \sin x \cos y dy = 0$ is :
• Q58: The solution of $\frac{dy}{dx} + \frac{y}{x} = e^x$ is:
• Q59: The differential equation of the family of curves $x^2 + y^2 – 2ay = 0$, where $a$ is arbitrary constant, is:
• Q60: Family $y = Ax + A^3$ of curves will correspond to a differential equation of order
• Q61: The general solution of $\frac{dy}{dx} = e^{y-2x}$ is :
• 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 :
• Q63: The general solution of the differential equation $\frac{dy}{dx} = e^{x^2} + 2xy$ is :
• Q64: The solution of the equation $(2y – 1) dx – (2x + 3)dy = 0$ is :
• Q65: The differential equation for which $y = a\cos x + b\sin x$ is a solution, is :
• Q66: The solution of $\frac{dy}{dx} + y = e^{-x}, y (0) = 0$ is :
• Q67: The order and degree of the differential equation $(\frac{d^3y}{dx^3})^2 - 3(\frac{d^2y}{dx^2}) + 2(\frac{dy}{dx})^4 = y$ are :
• Q68: The order and degree of the differential equation $[1 + (\frac{dy}{dx})^2]^{\frac{3}{2}} = \frac{d^2y}{dx^2}$ are :
• Q69: The differential equation of the family of curves $y^2 = 4a (x + a)$ is :
• Q70: Which of the following is the general solution of $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$ ?
• Q71: General solution of $\frac{dy}{dx} + y \tan x = \sec x$ is :
• Q72: Solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \sin x$ is :
• Q73: The general solution of the differential equation $(e^x + 1) ydy = (y + 1) e^x dx$ is:
• Q74: The solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}$ is :
• Q75: The solution of the differential equation $\frac{dy}{dx} + \frac{2xy}{1+x^2} = \frac{1}{(1+x^2)^2}$ is :
• Q76(i): Fill in the blanks: The degree of the differential equation $\frac{d^2y}{dx^2} + e^{\frac{dy}{dx}} = 0$ is _________.
• Q76(ii): Fill in the blanks: The degree of the differential equation $\sqrt{1 + (\frac{dy}{dx})^2} = x$ is _________.
• Q76(iii): Fill in the blanks: The number of arbitrary constants in the general solution of a differential equation of order three is _________.
• Q76(iv): Fill in the blanks: $\frac{dy}{dx} + \frac{y}{x\log x} = \frac{1}{x}$ is an equation of the type _________.
• Q76(ix): Fill in the blanks: General solution of $\frac{dy}{dx} + \frac{y}{x} = \sin x$ is _________.
• Q76(v): Fill in the blanks: General solution of the differential equation of the type $\frac{dx}{dy} + P_1 x = Q_1$ is given by _________.
• Q76(vi): Fill in the blanks: The solution of the differential equation $x\frac{dy}{dx} + 2y = x^2$ is _________.
• Q76(vii): Fill in the blanks: The solution of $(1 + x^2) \frac{dy}{dx} +2xy – 4x^2 = 0$ is _________.
• Q76(viii): Fill in the blanks: The solution of the differential equation $ydx + (x + xy)dy = 0$ is ______.
• Q76(x): Fill in the blanks: The solution of differential equation $\cot y dx = xdy$ is _________.
• Q76(xi): Fill in the blanks: The integrating factor of $\frac{dy}{dx} + \frac{y}{x(y+1)} = 1$ is _________.
• Q77(i): State True or False: Integrating factor of the differential of the form $\frac{dx}{dy} + p_1 x = Q_1$ is given by $e^{\int p_1 dy}$.
• Q77(ii): State True or False: Solution of the differential equation of the type $\frac{dx}{dy} + p_1 x = Q_1$ is given by $x \cdot I.F. = \int (Q_1 \times I.F.) dy$.
• Q77(iii): State True or False: Correct substitution for the solution of the differential equation of the type $\frac{dy}{dx} = f(x, y)$, where $f (x, y)$ is a homogeneous function of zero degree is $y = vx$.
• Q77(iv): State True or False: Correct substitution for the solution of the differential equation of the type $\frac{dx}{dy} = g(x, y)$ where $g(x, y)$ is a homogeneous function of the degree zero is $x = vy$.
• Q77(ix): State True or False: The solution of the differential equation $\frac{dy}{dx} = \frac{x+2y}{x}$ is $x + y = kx^2$.
• Q77(vi): State True or False: The differential equation representing the family of circles $x^2 + (y – a)^2 = a^2$ will be of order two.
• Q77(vii): State True or False: The solution of $\frac{dy}{dx} = (\frac{y}{x})^{\frac{1}{3}}$ is $y^{\frac{2}{3}} – x^{\frac{2}{3}} = c$.
• Q77(viii): State True or False: Differential equation representing the family of curves $y = e^x (A\cos x + B\sin x)$ is $\frac{d^2y}{dx^2} – 2\frac{dy}{dx} + 2y = 0$.
• Q77(x): State True or False: Solution of $\frac{dy}{dx} = \frac{y}{x} + \tan(\frac{y}{x})$ is $\sin(\frac{y}{x}) = cx$.
• Q77(xi): State True or False: The differential equation of all non horizontal lines in a plane is $\frac{d^2x}{dy^2} = 0$.