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Q76(xi):
Fill in the blanks: The integrating factor of $\frac{dy}{dx} + \frac{y}{x(y+1)} = 1$ is _________.
Solution :
Step 1: Analyis of the Differential Equation
The given equation is $\frac{dy}{dx} + \frac{y}{x(y+1)} = 1$. Rearranging the equation by treating $x$ as a function of $y$ ($x = f(y)$), we invert the derivative using the property $\frac{dx}{dy} = \frac{1}{dy/dx}$.
Starting with: $\frac{dy}{dx} = 1 - \frac{y}{x(y+1)}$ $\frac{dy}{dx} = \frac{x(y+1) - y}{x(y+1)}$
Inverting both sides: $\frac{dx}{dy} = \frac{x(y+1)}{x(y+1) - y}$ $\frac{dx}{dy} = \frac{x(y+1)}{xy + x - y}$
Step 2: Transforming into Linear Form
Divide the numerator and denominator by $x$: $\frac{dx}{dy} = \frac{y+1}{y + 1 - \frac{y}{x}}$ [This does not immediately yield a linear form in $x$].
Let us re-examine the original equation: $\frac{dy}{dx} + \frac{y}{x(y+1)} = 1$. Multiply throughout by $(y+1)$: $(y+1)\frac{dy}{dx} + \frac{y}{x} = y+1$
Let $u = y^2 + 2y$ (not suitable here). Let us consider the substitution $v = y^2$. This is a Bernoulli-type transformation. However, if we rewrite as: $\frac{dx}{dy} = \frac{x(y+1)}{xy+x-y}$ $xy+x-y = x(y+1)\frac{dx}{dy}$ $x(y+1) - y = x(y+1)\frac{dx}{dy}$
Rearranging into the standard linear form $\frac{dx}{dy} + P(y)x = Q(y)$: $x(y+1)\frac{dx}{dy} - x(y+1) = -y$ $\frac{dx}{dy} - x = \frac{-y}{y+1}$ [Per linear differential equation structure].
Step 3: Calculating the Integrating Factor (IF)
The standard linear form is $\frac{dx}{dy} + P(y)x = Q(y)$. Here, $P(y) = -1$.
The formula for the Integrating Factor is: $IF = e^{\int P(y) \, dy}$ [By the definition of integrating factors for linear differential equations].
Substituting $P(y) = -1$: $IF = e^{\int -1 \, dy}$ $IF = e^{-y}$
Step 4: Verification via SVG Representation
Final Solution: The integrating factor is $e^{-y}$
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 _________.
- 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(v): State True or False: Number of arbitrary constants in the particular solution of a differential equation of order two is two.
- 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 $\frac{d^2x}{dy^2} = 0$.
CBSE Solutions for Class 12 Mathematics Differential Equations
Chapters in CBSE - Class 12 Mathematics
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