Differential Equations

33 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Differential Equations' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Differential Equations'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Differential Equations'? medium MCQ Problem Solving & Optimization
• 4 Solve the second‑order linear homogeneous differential equation y''-5y'+6y=0 and determine the solution that satisfies the initial conditions y(0)=2 and y'(0)=0. hard 3 marks Characteristic equation for constant coefficient ODE
• 5 Which method is used to solve a first‑order linear differential equation of the form dy/dx+P(x)y=Q(x)? easy MCQ 1 mark Solution technique for linear ODE
• 6 Solve the linear first‑order differential equation dy/dx+2y=4x and find the particular solution that satisfies y(0)=1. medium 2 marks Integrating factor for linear ODE
• 7 Find the general solution of the differential equation dy/dx=3x^2 and state its domain. easy 1 mark First‑order separable differential equation
• 8 Solve the separable differential equation dy/dx=x^2/y and state the domain of the solution. easy 2 marks Separable differential equation
• 9 Which method is appropriate for solving the first‑order differential equation dy/dx+y = y^2e^x? medium MCQ 2 marks Bernoulli differential equation
• 10 Given the differential equation (2xy + y^2)dx + (x^2 + 2xy)dy = 0, (i) show that it is not exact, (ii) find an integrating factor of the form μ(x), and (iii) obtain the implicit solution. hard 5 marks Exact differential equations and integrating factor
• 11 Solve the non‑homogeneous linear differential equation y''-3y'+2y = e^2x and find the particular solution satisfying the initial conditions y(0)=1 and y'(0)=0. hard 5 marks Second‑order linear ODE with constant coefficients (non‑homogeneous)
• 12 Given the differential equation (y e^xy+x) dx + (x e^xy+y) dy = 0: (i) show that it is not exact, (ii) find an integrating factor that depends only on y, and (iii) obtain the implicit solution. hard 5 marks Integrating factor for non‑exact equation
• 13 Solve the separable differential equation dy/dx=x^3/(1+y^2) and state the domain of the solution. easy 2 marks Separable differential equation
• 14 Solve the second‑order linear differential equation y''+4y=sin 2x with the initial conditions y(0)=0 and y'(0)=1. hard 4 marks Second‑order linear non‑homogeneous ODE
• 15 Given the differential equation (y+2x) dx+(x+3y^2) dy=0, (i) show that it is not exact, (ii) find an integrating factor that depends only on x, and (iii) obtain the implicit solution. hard 4 marks Exact equations and integrating factors
• 16 Solve the second‑order linear differential equation y''-4y'+4y = e^2x with the initial conditions y(0)=3 and y'(0)=5. hard 4 marks Second‑order linear ODE with repeated root
• 17 Solve the separable differential equation dy/dx=2x/(1-y^2) and state the maximal interval on which the solution is defined, given the initial condition y(0)=0. easy 2 marks Separable differential equation
• 18 Solve the separable differential equation dy/dx=2x/(1+y^2) with the initial condition y(0)=1. State the interval on which the solution is valid. medium 2 marks Separable ODE and arctan integration
• 19 Solve the non‑homogeneous linear differential equation y''-3y'+2y=e^x with the initial conditions y(0)=0 and y'(0)=1. hard 5 marks Second‑order linear ODE with constant coeffs
• 20 Given the differential equation (2xy + y^2) dx + (x^2+2xy) dy =0, (i) show that it is not exact, (ii) find an integrating factor of the form μ(x), and (iii) obtain the implicit solution. hard 5 marks Exactness and integrating factor $\mu(x)$
• 21 Which method is appropriate for solving the first‑order differential equation dy/dx+y=sin x? easy MCQ 1 mark Identify linear first‑order ODE
• 22 Consider the differential equation (3x^2y+y^3) dx+(x^3+3xy^2) dy=0. (i) Show that it is not exact. (ii) Find an integrating factor that depends only on y. (iii) Obtain the implicit solution. hard 4 marks Exact equations and integrating factor
• 23 Solve the separable differential equation dy/dx=x^2/(1-y^2) with the initial condition y(0)=0 and state the interval where the solution is valid. medium 2 marks Separable ODE with initial condition
• 24 Solve the second‑order linear differential equation y''+y=4cos 2x with the initial conditions y(0)=2 and y'(0)=-1. hard 3 marks Second‑order linear ODE with constant coefficients
• 25 Solve the second‑order linear differential equation y''-2y'+y = e^x with the initial conditions y(0)=1 and y'(0)=0. medium 4 marks Second‑order linear ODE with repeated root
• 26 Which method is most appropriate for solving the first‑order differential equation dydx-3y=6x? easy MCQ 1 mark Linear first‑order ODE
• 27 Solve the separable differential equation dydx=(x^2+1)/y^2 with the initial condition y(1)=2, and state the interval on which the solution is valid. medium 3 marks Separable differential equation
• 28 Given the differential equation (ycos x + xsin y) dx + (xcos y - ysin x) dy =0: (i) Show that it is not exact, (ii) Find an integrating factor that depends only on x, (iii) Obtain the implicit solution. hard 5 marks Integrating factor depending on x
• 29 Solve the second‑order linear differential equation y''-6y'+9y = e^3x with the initial conditions y(0)=2 and y'(0) = -1. hard 5 marks Second‑order linear ODE with repeated root
• 30 Consider the differential equation (xy^2+y) dx+(x^2y+x) dy=0. (i) Show that it is not exact. (ii) Find an integrating factor that depends only on x. (iii) Obtain the implicit solution. medium 3 marks Exact equation with integrating factor
• 31 Solve the separable differential equation dy/dx=3x^2/2y with the initial condition y(0)=4. State the interval on which the solution is valid. easy 2 marks Separable first‑order ODE
• 32 For the differential equation (2xy + y^2)dx + (x^2 + 2xy)dy = 0, find an integrating factor that depends only on x which makes the equation exact. medium 2 marks Integrating factor for first-order DE
• 33 Solve the second‑order linear differential equation y''-4y'+5y = e^2x with the initial conditions y(0)=1 and y'(0)=0. hard 4 marks Second‑order linear differential equation