NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations

Calculus helps students understand how mathematics describes change and real-life phenomena, and Chapter 9: Differential Equations is a conceptually important and scoring chapter in Class 12 Maths. This chapter introduces students to the formation and solution of differential equations, which are widely used to model real-world situations in physics, biology, and economics.

NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations are prepared strictly as per the CBSE syllabus and exam pattern. These solutions include important CBSE board questions asked between 2018 and 2025 and are explained in a clear, step-by-step manner using simple language and proper mathematical presentation. This helps students develop strong conceptual clarity, practise effectively, and score well in board examinations.

NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations

Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differentialequation:y + 5y=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential} { } {equation:} \\ {y\ +\ 5y} { } {=} { } {0} 
Differential Equations
Medium
Q.
 Verify that the given functions (explicit or implicit)is a solution of the corresponding differential equation:(y = e)x+ 1:  y – y = 0{Verify\ that\ the\ given\ functions\ } \left(\right. {explicit\ or\ implicit} \left.\right) {is\ a\ solution\ of\ the\ corresponding\ differential\ equation:} \\ \left({y\ =\ e}\right)^{{x}} {+\ 1} { } { } { } { } { } {:\ } { } { } {\ y\ –\ y\ =\ 0}  
Differential Equations
Difficult
Q.
For the differential equation given below, find the general solution:
dy dx =si n -1 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8MrFz0xbbf9v8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeqabaWaaqaafaaake aadaWcaaqaaGqabiaa=rgacaWF5baabaGaa8hzaiaa=HhaaaGaeyyp a0Jaa83Caiaa=LgacaWFUbWaaWbaaSqabeaacaWFTaGaa8xmaaaaki aa=Hhaaaa@434F@
Differential Equations
Medium
Q.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Differential Equations
Easy
Q.
Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
x a + y b =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacH8Mrpy0xbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeqabaWaaqaafaaake aadaWcaaqaaGqabiaa=HhaaeaacaWFHbaaaiaa=TcadaWcaaqaaiaa =LhaaeaacaWFIbaaaiaa=1dacaWFXaaaaa@3EBE@
Differential Equations
Medium
Q.
Verify that the given functions ( explicit or implicit )is a solution of the corresponding differential equation: y = x sinx     :  xy' = y + x  x 2  – y 2   ( x≠0 and x > y or x < –y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVD0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaabAfacaqGLbGaaeOCaiaabMgacaqGMbGaaeyEaiaabcca caqG0bGaaeiAaiaabggacaqG0bGaaeiiaiaabshacaqGObGaaeyzai aabccacaqGNbGaaeyAaiaabAhacaqGLbGaaeOBaiaabccacaqGMbGa aeyDaiaab6gacaqGJbGaaeiDaiaabMgacaqGVbGaaeOBaiaabohaca qGGaWdamaabmaabaWdbiaabwgacaqG4bGaaeiCaiaabYgacaqGPbGa ae4yaiaabMgacaqG0bGaaeiiaiaab+gacaqGYbGaaeiiaiaabMgaca qGTbGaaeiCaiaabYgacaqGPbGaae4yaiaabMgacaqG0baapaGaayjk aiaawMcaa8qacaqGPbGaae4CaiaabccacaqGHbGaaeiiaiaabohaca qGVbGaaeiBaiaabwhacaqG0bGaaeyAaiaab+gacaqGUbGaaeiiaiaa b+gacaqGMbGaaeiiaiaabshacaqGObGaaeyzaiaabccacaqGJbGaae 4BaiaabkhacaqGYbGaaeyzaiaabohacaqGWbGaae4Baiaab6gacaqG KbGaaeyAaiaab6gacaqGNbGaaeiiaiaabsgacaqGPbGaaeOzaiaabA gacaqGLbGaaeOCaiaabwgacaqGUbGaaeiDaiaabMgacaqGHbGaaeiB aiaabccacaqGLbGaaeyCaiaabwhacaqGHbGaaeiDaiaabMgacaqGVb GaaeOBaiaabQdaaeaapaGaaeyEaiaabccacaqG9aGaaeiiaiaabIha caqGGaGaae4CaiaabMgacaqGUbGaaeiEaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaeOoaiaabccacaqGGaGaaeiEaiaabMhacaqGNaGa aeiiaiaab2dacaqGGaGaaeyEaiaabccacaqGRaGaaeiiaiaabIhaca qGGaWaaOaaaeaacaqG4bWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaa bobicaqGGaGaaeyEamaaCaaaleqabaGaiGgGbkdaaaaabeaakiaayk W7caaMc8+aaeWaaeaacaqG4bGaeyiyIKRaaeimaiaaykW7caqGHbGa aeOBaiaabsgacaqGGaGaaeiEaiaabccacaqG+aGaaeiiaiaabMhaca qGGaGaae4BaiaabkhacaqGGaGaaeiEaiaabccacaqG8aGaaeiiaiaa bobicaqG5baacaGLOaGaayzkaaaaaaa@D43D@
Differential Equations
Medium
Q.
 Verify that the given functions (explicit or implicit)is a solution of the corresponding differential equation:y = Ax:  xy = y(x ≠ 0){Verify\ that\ the\ given\ functions\ } \left(\right. {explicit\ or\ implicit} \left.\right) {is\ a\ solution\ of\ the\ corresponding\ differential\ equation:} \\ {y\ =\ Ax} { } { } { } { } { } { } { } { } { } { } { } {:\ } { } { } {\ xy\ =\ y} { } { } \left(\right. {x\ } \neq {\ 0} \left.\right)  
Differential Equations
Easy
Q.
 Verify that the given functions (explicit or implicit)is a solution of the corresponding differential equation:y = cos x + C:  y + sinx = 0{Verify\ that\ the\ given\ functions\ } \left(\right. {explicit\ or\ implicit} \left.\right) {is\ a\ solution\ of\ the\ corresponding\ differential\ equation:} \\ {y\ =\ cos\ x\ +\ C} { } { } { } { } { } { } {:\ } { } { } {\ y\ +\ sinx\ =\ 0}  
Differential Equations
Easy
Q.
 Verify that the given functions (explicit or implicit)is a solution of the corresponding differential equation:(y = x)2+2x + C:  y –2x –2 = 0{Verify\ that\ the\ given\ functions\ } \left(\right. {explicit\ or\ implicit} \left.\right) {is\ a\ solution\ of\ the\ corresponding\ differential\ equation:} \\ \left({y\ =\ x}\right)^{{2}} {+2x\ +\ C} { } { } { } { } {:\ } { } { } {\ y\ –2x\ –2\ =\ 0}  
Differential Equations
Easy
Q.
 The order of the differential equation(2x)2(d)2y(dx)2–3dydx+y=0is(A)3  (B) 2    (C) 0    (D) not​ defined{The\ order\ of\ the\ differential\ equation} \\ \left({2x}\right)^{{2}} \frac{\left({d}\right)^{{2}} {y}}{\left({dx}\right)^{{2}}} { } {–} { } {3} \frac{{dy}}{{dx}} { } {+} { } {y} { } {=} { } {0} { } { } {is} \\ \left(\right. {A} \left.\right) {3 \ \ } { } { } \left(\right. {B} )\ {2 \ \ \ \ } { } { } \left(\right. {C} )\ {0 \ \ \ \ } { } { } \left(\right. {D} \left.\right) {\ not} {​} {\ defined}  
Differential Equations
Medium
Q.
 Determine order and degree (if defined) of differentialequation:(dsdt)4+ 3sd2sdt2=0 \begin{array}{l}{Determine\ order\ and\ degree\ }\left({if\ defined}\right){\ of\ differential}{\hspace{0.33em}}{equation:} \\ {\left(\frac{{ds}}{{dt}}\right)}^{{4}}{+\ 3s}\frac{{{d}}^{{2}}{s}}{{{dt}}^{{2}}}{=0} \end{array} 
 
Differential Equations
Easy
Q.
 The degree of the differential equation(((d)2y(dx)2))3+((dydx))2+ sin(dydx)+1=0is(A) 3   (B) 2       (C) 1        (D) not​ defined{The\ degree\ of\ the\ differential\ equation} \\ \left(\left(\right. \frac{\left({d}\right)^{{2}} {y}}{\left({dx}\right)^{{2}}} \left.\right)\right)^{{3}} {+} \left(\left(\right. \frac{{dy}}{{dx}} \left.\right)\right)^{{2}} {+\ sin} \left(\right. \frac{{dy}}{{dx}} \left.\right) {+1} { } {=} { } {0} { } { } {is} \\ \left(\right. {A} \left.\right) {\ 3 \ \ \ } { } { } \left(\right. {B} )\ {2} { \ \ \ \ \ \ \ } { } { } \left(\right. {C} \left.\right) {\ 1} {\ \ \ \ \ \ \ \ } { } \left(\right. {D} )\ { } {not} {​} {\ defined}  
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differential equation:y+ 2y+ siny=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential\ equation:} \\ {y} { } {+\ 2y} { } {+\ siny} { } {=} { } {0} 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differential equation:y + ((y))2+2y=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential\ equation:} \\ {y\ +\ } \left(\left({y}\right)\right)^{{2}} {+} { } {2y} { } {=} { } {0} 
 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differential equation:y+ y=(e)x{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential\ equation:} \\ {y} { } {+\ y} { } {=} { } \left({e}\right)^{{x}} 
 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differential equation:y + 2y+y=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential\ equation:} \\ {y\ +\ 2y} { } {+} { } {y} { } {=} { } {0} 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differential equation:((y))2 + ((y))3+((y))4+((y))5=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential\ equation:} \\ \left(\left({y}\right)\right)^{{2}} {\ +\ } \left(\left({y}\right)\right)^{{3}} {+} \left(\left({y}\right)\right)^{{4}} {+} \left(\left({y}\right)\right)^{{5}} {=} { } {0} 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differentialequation:(d)2y(dx)2=cos3x + sin3x{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential} { } {equation:} \\ \frac{\left({d}\right)^{{2}} {y}}{\left({dx}\right)^{{2}}} { } {=} { } {cos3x\ +\ sin3x} 
Differential Equations
Easy
Q.
 Determine order and degree (if defined) of differentialequation:(((d)2y(dx)2))2+cos(dydx)=0{Determine\ order\ and\ degree\ } \left({if\ defined}\right) {\ of\ differential} { } {equation:} \\ \left(\left(\frac{\left({d}\right)^{{2}} {y}}{\left({dx}\right)^{{2}}}\right)\right)^{{2}} {+} { } {cos} \left(\frac{{dy}}{{dx}}\right) {=} { } {0} 
Differential Equations
Medium
Q.
Solve  the  differential  equation  y exydx=(xexy+y2)dy  (y≠0).

Q1. Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

Answer: Area = 14/3 sq units

Q2. Find the area of the smaller region bounded by the ellipse x²/9 + y²/4 = 1 and the line x/3 + y/2 = 1.

Answer: Area = 3π/2 − 3 sq units

Q3. Find the area bounded by the curves y = x² and y = |x|.

Answer: Area = 1/3 sq unit

Q4. Draw the rough sketch of the region enclosed by y = x and y = x² and find its area.

Answer: Area = 1/6 sq unit

Q5. Find the area of the region bounded by the lines 2x + y = 4, 3x − 2y = 6 and x − 3y + 5 = 0.

Answer: Area = 2 sq units

Q6. Find the area of the region { (x, y) : y² ≥ 4x, 4x² + 4y² ≤ 9 }.

Answer: Area = 9π/8 − 3/2 sq units

Q7. Find the area of the region in the first quadrant enclosed by x-axis, line x = √3y and the circle x² + y² = 4.

Answer: Area = π/3 sq units

Q8. Area lying in the first quadrant and bounded by the circle x² + y² = 4 and lines x = 0 and x = 2.

Answer: Area = π sq units

Q9. Area lying between the curves y² = 4x and y = 2x.

Answer: Area = 1/3 sq unit

Q10. Find the area bounded by the curve y = sin x between x = 0 and x = 2π.

Answer: Area = 4 sq units

Q11. Sketch y = |x + 3| and evaluate ∫−60 |x + 3| dx.

Answer: Area = 9 sq units

Q12. Find the area enclosed between the parabola y² = 4ax and the line y = mx.

Answer: Area = 8a³ / (3m³)

Q13. Area bounded by y = x³, x-axis and x = −2, x = 1.

Answer: Area = 15/4 sq units

Q14. Area bounded by y = x|x|, x-axis and x = −1, x = 1.

Answer: Area = 2/3 sq unit

Q15. Area of the circle x² + y² = 16 exterior to the parabola y² = 6x.

Answer: Area = (4/3)(4π − √3)

Q16. Area bounded by y-axis, y = cos x and y = sin x for 0 ≤ x ≤ π/2.

Answer: Area = 2(√2 − 1)

Q17. Area bounded by y² = 9x, x = 2, x = 4 and x-axis in first quadrant.

Answer: Area = 14 sq units

Q18. Area bounded by x² = 4y, y = 2, y = 4 and y-axis.

Answer: Area = 16/3 sq units

Q19. Find the area of the ellipse x²/16 + y²/9 = 1.

Answer: Area = 12π sq units

Q20. Find the area of the ellipse x²/4 + y²/9 = 1.

Answer: Area = 6π sq units

FAQs: NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations

Q1. Why is Differential Equations important for Class 12 Maths?

This chapter forms the foundation of advanced calculus and carries good weightage in CBSE board exams, especially for long-answer questions.

Q2. How many marks are usually asked from Chapter 9 in board exams?

Generally, 5 to 8 marks worth of questions are asked from Differential Equations in the CBSE board exams.

Q3. Are NCERT Solutions sufficient for board exam preparation?

Yes. NCERT textbook exercises and NCERT Solutions are sufficient for CBSE board exams, as most questions are directly based on NCERT problems.

Q4. Which topics are most important in this chapter?

The most important topics include:

  • Formation of differential equations

  • Solving differential equations

  • Linear differential equations

Q5. Why do students find Differential Equations difficult?

Students often face difficulty in:

  • Identifying the correct type of differential equation

  • Choosing the appropriate solving method

  • Avoiding calculation and sign errors

Regular practice and step-by-step solving help overcome these issues.

Q6. Is Differential Equations useful for competitive exams?

Yes. This chapter is important for JEE and other entrance exams, as differential equations are frequently used in calculus-based problems.