NCERT Exemplar Solutions Class 12 Maths Differential Equations — Fill in the Blanks and True or False Q76–77

NCERT Exemplar Problems, Class 12 Mathematics — Differential Equations, Question 76 (fill in the blanks, parts i to xi) and Question 77 (true or false, parts i to xi). All 22 parts solved.

Each part of Q76 and Q77 has its own heading and its own answer box. For Q76 the box holds the missing expression or word; for Q77 the box holds just True or False, with the reason given in the solution above it.

A note on the syllabus. The rationalised NCERT Chapter 9 keeps order and degree (Exercise 9.1), general and particular solutions (Exercise 9.2), and solving first-order first-degree equations: variables separable (9.3), homogeneous (9.4) and linear (9.5). It dropped formation of a differential equation from a given family of curves. JEE Main 2026 lists order and degree, separation of variables, homogeneous and linear equations, and omits formation. JEE Advanced 2026 lists “formation of ordinary differential equations”. On this page, Questions 77(vi), 77(viii) and 77(xi) are formation questions, so they matter for JEE Advanced but not for the rationalised textbook or JEE Main. Everything else is in syllabus. Q76(i) and Q76(ii) use the usual degree convention: clear radicals and fractional powers of derivatives first, then read the power of the highest-order derivative; the degree is “not defined” only when a derivative sits inside a transcendental function.

Key insight. Most of these parts are classification or recall: name the type (linear, separable, homogeneous), recall its integrating factor or substitution, and check that the stated solution actually satisfies the equation. For true or false, a quick differentiation of the claimed solution settles it.

Question 76(i)

The degree of the differential equation $\dfrac{d^2y}{dx^2}+e^{\frac{dy}{dx}}=0$ is ______.

Solution. The derivative $\dfrac{dy}{dx}$ sits inside the exponential $e^{(\cdot)}$, a transcendental function, and the equation cannot be written as a polynomial in the derivatives. Degree is defined only for such polynomial equations, so it is not defined. (The order is $2$.)

Question 76(ii)

The degree of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=x$ is ______.

Solution. Remove the radical first by squaring both sides: $1+\left(\dfrac{dy}{dx}\right)^2=x^2$. This is a polynomial in $\dfrac{dy}{dx}$, and the highest power of the highest-order derivative is $2$. The degree is $2$.

A note on the printed answer. The key prints “not defined”, reading any radical as making the degree undefined. We follow the JEE and standard textbook convention: clear radicals and fractional powers of derivatives first, then read the power. Under that convention the degree is $2$.

Question 76(iii)

The number of arbitrary constants in the general solution of a differential equation of order three is ______.

Solution. The general solution of an $n$th order equation carries exactly $n$ independent arbitrary constants, one for each integration. For order three, that is $3$.

Question 76(iv)

$\dfrac{dy}{dx}+\dfrac{y}{x\log x}=\dfrac1x$ is an equation of the type ______.

Solution. The unknown $y$ and its derivative appear only to the first power, with coefficients that depend on $x$ alone, so the equation has the form $\dfrac{dy}{dx}+Py=Q$ with $P=\dfrac1{x\log x}$ and $Q=\dfrac1x$. It is a linear differential equation of the first order.

Question 76(v)

The general solution of the differential equation of the type $\dfrac{dx}{dy}+P_1x=Q_1$ is given by ______.

Solution. Here $x$ is the dependent variable, so the integrating factor is $e^{\int P_1\,dy}$. Multiplying through makes the left side $\dfrac{d}{dy}\left(x\,e^{\int P_1dy}\right)$, and integrating with respect to $y$ gives the solution.

Question 76(vi)

The solution of the differential equation $x\dfrac{dy}{dx}+2y=x^2$ is ______.

Solution. This is linear. Divide by $x$: $\dfrac{dy}{dx}+\dfrac2x y=x$. The integrating factor is $e^{\int 2/x\,dx}=x^2$. Then $\dfrac{d}{dx}(x^2y)=x^3$, so $x^2y=\dfrac{x^4}{4}+c$.

Question 76(vii)

The solution of $(1+x^2)\dfrac{dy}{dx}+2xy-4x^2=0$ is ______.

Solution. Rewrite as the linear equation $\dfrac{dy}{dx}+\dfrac{2x}{1+x^2}y=\dfrac{4x^2}{1+x^2}$. The integrating factor is $e^{\log(1+x^2)}=1+x^2$, so $\dfrac{d}{dx}\big[(1+x^2)y\big]=4x^2$. Integrating, $(1+x^2)y=\dfrac{4x^3}{3}+\dfrac c3$, that is, $3y(1+x^2)=4x^3+c$ (the constant has been rescaled).

Question 76(viii)

The solution of the differential equation $y\,dx+(x+xy)\,dy=0$ is ______.

Solution. Variables separable. Write $y\,dx=-x(1+y)\,dy$ and divide by $xy$: $\dfrac{dx}{x}=-\dfrac{1+y}{y}\,dy=-\left(\dfrac1y+1\right)dy$. Integrating, $\log|x|=-\log|y|-y+c$, so $\log|xy|+y=c$. Taking exponentials, $xy=Ae^{-y}$.

Question 76(ix)

The general solution of $\dfrac{dy}{dx}+y=\sin x$ is ______.

Solution. Linear with integrating factor $e^{x}$. Then $\dfrac{d}{dx}(ye^x)=e^x\sin x$, and $\int e^x\sin x\,dx=\dfrac{e^x(\sin x-\cos x)}{2}$. Hence $ye^x=\dfrac{e^x(\sin x-\cos x)}{2}+c$.

Question 76(x)

The solution of the differential equation $\cot y\,dx=x\,dy$ is ______.

Solution. Separate the variables: $\dfrac{dx}{x}=\tan y\,dy$. Integrating, $\log|x|=-\log|\cos y|+\log|c|$, so $x\cos y=c$, that is, $x=c\sec y$.

Question 76(xi)

The integrating factor of $\dfrac{dy}{dx}+y=\dfrac{1+y}{x}$ is ______.

Solution. Collect the $y$ terms: $\dfrac{dy}{dx}+\left(1-\dfrac1x\right)y=\dfrac1x$. So $P=1-\dfrac1x$ and the integrating factor is $e^{\int P\,dx}=e^{x-\log x}=\dfrac{e^x}{x}$.

Question 77(i)

Integrating factor of the differential of the form $\dfrac{dx}{dy}+p_1x=Q_1$ is given by $e^{\int p_1\,dy}$.

Solution. With $x$ as the dependent variable, the integrating factor is the exponential of the integral of the coefficient with respect to $y$. The statement is true.

Question 77(ii)

Solution of the differential equation of the type $\dfrac{dx}{dy}+p_1x=Q_1$ is given by $x\cdot\text{I.F.}=\int(\text{I.F.})\times Q_1\,dy$.

Solution. Multiplying by the integrating factor turns the left side into $\dfrac{d}{dy}(x\cdot\text{I.F.})$, and integrating gives the stated form. The statement is true, with the understanding that the arbitrary constant $+C$ is added on the right.

Question 77(iii)

Correct substitution for the solution of the differential equation of the type $\dfrac{dy}{dx}=f(x,y)$, where $f(x,y)$ is a homogeneous function of zero degree, is $y=vx$.

Solution. When $f$ is homogeneous of degree zero, $f(x,y)=f(1,y/x)$, so $y=vx$ gives $v+x\dfrac{dv}{dx}=f(1,v)$, which separates. True.

Question 77(iv)

Correct substitution for the solution of the differential equation of the type $\dfrac{dx}{dy}=g(x,y)$, where $g(x,y)$ is a homogeneous function of the degree zero, is $x=vy$.

Solution. Here $x$ is the dependent variable, so the roles swap: $x=vy$ gives $v+y\dfrac{dv}{dy}=g(v,1)$, which separates. True.

Question 77(v)

Number of arbitrary constants in the particular solution of a differential equation of order two is two.

Solution. A particular solution is obtained by fixing the arbitrary constants using given conditions, so it contains no arbitrary constants at all. It is the general solution of an order-two equation that has two.

Question 77(vi)

The differential equation representing the family of circles $x^2+(y-a)^2=a^2$ will be of order two.

Solution. The family has a single parameter $a$ (centre $(0,a)$ on the $y$-axis, touching the origin), so one differentiation eliminates it and the equation is of order one. Expanding, $x^2+y^2=2ay$. Differentiating, $2x+2yy’=2ay’$, and substituting $a=\dfrac{x^2+y^2}{2y}$ gives $(x^2-y^2)\,y’=2xy$, which is first order.

Question 77(vii)

The solution of $\dfrac{dy}{dx}=\left(\dfrac yx\right)^{\frac13}$ is $y^{\frac23}-x^{\frac23}=c$.

Solution. Separate: $y^{-1/3}dy=x^{-1/3}dx$. Integrating, $\dfrac32y^{2/3}=\dfrac32x^{2/3}+c_1$, so $y^{2/3}-x^{2/3}=c$ with $c=\tfrac23c_1$. True.

Question 77(viii)

Differential equation representing the family of curves $y=e^x(A\cos x+B\sin x)$ is $\dfrac{d^2y}{dx^2}-2\dfrac{dy}{dx}+2y=0$.

Solution. This is a formation question. Two constants means two differentiations. Writing $y=e^x u$ with $u=A\cos x+B\sin x$ gives $y’=y+e^xu’$ and $y”=y’+e^x(u’+u”)$ with $u”=-u$. Hence $y”=2y’-2y$, that is, $y”-2y’+2y=0$. True.

Question 77(ix)

The solution of the differential equation $\dfrac{dy}{dx}=\dfrac{x+2y}{x}$ is $x+y=kx^2$.

Solution. Rewrite as the linear equation $\dfrac{dy}{dx}-\dfrac2xy=1$ with integrating factor $x^{-2}$. Then $\dfrac{d}{dx}\left(\dfrac y{x^2}\right)=\dfrac1{x^2}$, so $\dfrac{y}{x^2}=-\dfrac1x+k$, that is, $y=-x+kx^2$, or $x+y=kx^2$. True.

Question 77(x)

Solution of $x\dfrac{dy}{dx}=y+x\tan\dfrac yx$ is $\sin\left(\dfrac yx\right)=cx$.

Solution. Homogeneous: put $y=vx$, so $v+x\dfrac{dv}{dx}=v+\tan v$. Then $\cot v\,dv=\dfrac{dx}{x}$, giving $\log|\sin v|=\log|x|+\log|c|$, that is, $\sin\dfrac yx=cx$. True.

Question 77(xi)

The differential equation of all non horizontal lines in a plane is $\dfrac{d^2x}{dy^2}=0$.

Solution. A non-horizontal line can be written $x=my+c$, with two constants $m$ and $c$. Differentiating twice with respect to $y$ gives $\dfrac{d^2x}{dy^2}=0$. (Writing $y=mx+c$ would miss vertical lines, which is why $x$ is taken as the dependent variable.) This is another formation question. True.

Common mistakes

• Reading degree before clearing radicals. In Q76(ii) the radical hides a polynomial: squaring first gives degree $2$, not “not defined”.
• Calling a derivative inside $e^{(\cdot)}$ a power. In Q76(i) the term $e^{dy/dx}$ cannot be turned into a polynomial in $\dfrac{dy}{dx}$, so the degree really is undefined.
• Using $e^{\int P\,dx}$ when $x$ is the dependent variable. In Q76(v) and Q77(i), (ii) the integral is with respect to $y$.
• Counting constants in a particular solution. Q77(v): fixing constants from conditions leaves none, so the answer is False.
• Taking the number of constants from the shape of the curve. Q77(vi): the circles $x^2+(y-a)^2=a^2$ have one parameter, hence a first-order equation.
• Forgetting $+C$ or $x$ versus $y$ in formation. Q77(xi) uses $x=my+c$ so that vertical lines are included.

Practise next

• Other bands of this unit: short answer (Q1-24), long answer (Q25-33), objective, Q34-54 and objective, Q55-75.
• Textbook practice: Exercise 9.1, order and degree, Exercise 9.2, general and particular solutions, Exercise 9.3, variables separable, Exercise 9.4, homogeneous, Exercise 9.5, linear and the Miscellaneous Exercise.