Class 12 Maths · Chapter 9
Unit: Calculus
Tentative Study Share: 7.0 Marks
Chapter 9: Differential Equations — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Differential Equations.
NCERT Exercises in Chapter 9: Differential Equations
📘 NCERT Concept Notes — Differential Equations
01 · Read derivatives before classifying
Order is the highest derivative order present. Degree is defined only when the equation is polynomial in its derivatives. A trigonometric function of a derivative prevents assigning a degree.
02 · Separate without losing solutions
For a separable equation, move the y-factor with dy and x-factor with dx. Dividing by a factor can discard constant solutions; test its zeros in the original equation before finishing.
03 · Homogeneous means a ratio substitution
A first-order equation whose slope depends only on y/x is reduced by y=vx. This meaning of homogeneous is different from a linear equation with zero right-hand side.
04 · A linear equation needs standard form
Normalise the derivative coefficient to one. For y as a function of x, both P and Q must depend only on x. The integrating factor creates a product derivative. Initial conditions must lie in the original equation’s domain.
⚡ Key Formulas & Results — Differential Equations
Order and degree
What must be true before you state a degree?
\(\text{order}=\text{highest derivative order}\)
General and particular solutions
How does a particular solution differ from the general family?
\(F(x,y,C_1,\ldots,C_n)=0\)
Variable separation
What is the integral form of y′=f(x)g(y)?
\(\int\frac{dy}{g(y)}=\int f(x)dx+C\)
Derive proportional growth
How does y′=ky produce an exponential family?
\(y=Ce^{kx}\)
The ratio substitution
What must replace dy/dx when y=vx?
\(y=vx\quad\Rightarrow\quad\frac{dy}{dx}=v+x\frac{dv}{dx}\)
Derive the separable ratio equation
How does y′=F(y/x) become separable?
\(\frac{dv}{F(v)-v}=\frac{dx}{x}\)
Integrating factor and solution
What is the solution structure of y′+P(x)y=Q(x)?
\(\mu=e^{\int Pdx},\quad\mu y=\int\mu Qdx+C\)
Derive the integrating factor
Why does μ=e^{∫Pdx} turn the left side into a product derivative?
\((\mu y)\prime=\mu Q\)
When x depends on y
How do you solve dx/dy+P(y)x=Q(y)?
\(\mu(y)=e^{\int P(y)dy},\quad\mu x=\int\mu Q(y)dy+C\)
🎯 CBSE 2026 Board PYQs from Differential Equations
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q11
The integrating factor of differential equation R dx/dy + P x = Q, where P, Q, R are functions of y is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q12
The order and degree of the differential equation d/dx (eʸ) = 0 respectively are
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q29
Find the general solution of the differential equation x² dy/dx = x² + xy + y². OR Find the particular…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q4
The integrating factor of differential equation R dx/dy + P x = Q, where P, Q, R are functions of y is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q5
Find the order and degree of d(sin y)/dx=y².
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q27
Find the general solution of \(y^2\,dx+(x^2-xy+y^2)\,dy=0\). OR Solve \(dy/dx=y\tan x\) with \(y(0)=2\).
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q15
The integrating factor of differential equation R dx/dy + P x = Q, where P, Q, R are functions of y is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q16
Find the order and degree of \(\frac{d}{dx}(y^{\prime})^3+(y^{\prime})^3=1\).
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q30
Solve \((y^2-x^2)\,dx=2xy\,dy\). OR Solve \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), with \(y(1)=0\).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q11
Choose the general nonzero solution family for \(\frac{dy}{dx}=\frac{\sqrt y}{\sqrt x}\).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q12
Find the listed integrating factor for 2x y′−y=3.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q29
Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\). OR Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q4
Choose the listed general solution for x dy−y dx=0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q5
Find the listed integrating factor for 2x y′−y=3.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q27
Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\). OR Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q15
Choose the listed general solution for x²dy+y²dx=0.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q16
Find the listed integrating factor for 2x y′−y=3.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q30
Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\). OR Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).
📝 Chapterwise Practice Questions — Differential Equations (119 Questions)
Class 12 Differential Equations · Mcq Single Correct
Find the degree of x²y″ = (xy′ − y)³.
Class 12 Differential Equations · Mcq Single Correct
Find the degree of y″ + 3(y′)² = x² ln(y″).
Class 12 Differential Equations · Mcq Single Correct
Find the order and degree of [1 + (y′)²]² = y″.
Class 12 Differential Equations · Mcq Single Correct
The nonconstant solutions of 2xy′ − y = 3 lie on a family of which curves?
Class 12 Differential Equations · Mcq Single Correct
Find an integrating factor of (x ln x)y′ + y = 2 ln x, for x > 0, x ≠ 1.
Class 12 Differential Equations · Mcq Single Correct
Which function satisfies (y′)² − xy′ + y = 0?
Class 12 Differential Equations · Mcq Single Correct
Which function is not homogeneous in x and y?
Class 12 Differential Equations · Mcq Single Correct
Solve dx/x + dy/y = 0, with xy ≠ 0.
Class 12 Differential Equations · Mcq Single Correct
Find the general solution of xy′ + 2y = x², x ≠ 0.
Class 12 Differential Equations · Mcq Single Correct
Find the degree of sin x + cos(y′) = y².
Class 12 Differential Equations · Mcq Single Correct
Find the degree of [1 + (y′)²]^(3/2) = y″ after clearing the radical.
Class 12 Differential Equations · Mcq Single Correct
Find the order and degree of y″ + (y′)^(1/4) + x^(1/5) = 0 after clearing radicals in derivatives.
Class 12 Differential Equations · Mcq Single Correct
Solve y′ + 2y/x = 0 with y(1) = 1.
Class 12 Differential Equations · Mcq Single Correct
Find the general solution of y′ = e^(x−y) + x²e^(−y).
Class 12 Differential Equations · Mcq Single Correct
Find an integrating factor of (1 − y²)dx/dy + yx = ay, −1 < y < 1.
Class 12 Differential Equations · Mcq Single Correct
Solve y′ − y = 1 with y(0) = 1.
Class 12 Differential Equations · Mcq Single Correct
How many classical solutions of y′ = (y + 1)/(x − 1) satisfy y(1) = 2?
Class 12 Differential Equations · Mcq Single Correct
Find a real integrating factor of (1 − x²)y′ − xy = 1, for |x| < 1.
Class 12 Differential Equations · Mcq Single Correct
Find the product of the order and degree of y″sin y + (y′)³cos y = √y.
Class 12 Differential Equations · Mcq Single Correct
Find the sum of the order and degree of y″ + (y′)³ = sin y.
Class 12 Differential Equations · Mcq Single Correct
Solve x dy − (1 + x²)dx = dx, x ≠ 0.
Class 12 Differential Equations · Assertion Reason
Assertion (A): The degree of y″ = 1 + √(y′) is 2. Reason (R): Degree is the power of the highest…
Class 12 Differential Equations · Assertion Reason
Assertion (A): A solution of (1 + x²)y′ + y = arctan x is ye^(arctan x) = (arctan x − 1)e^(arctan x) +…
Class 12 Differential Equations · Assertion Reason
Assertion (A): An integrating factor of dx/dy + x tan y = sec²y is sec y. Reason (R): For dx/dy + Px =…