Differential Equations – important questions for Boards 2027
CBSE Class 12 Maths
For the differential equation (dy/dx)² + y = x, determine whether its degree is defined, and find it if defined.
Answer
The equation is a polynomial equation in its derivatives. Highest-order derivative: dy/dx ∴ Degree is defined and equals 2.
Which of the following is the general solution (primitive) of dy/dx = 2x? (A) y = x² + C (B) y = 2x + C (C) y = x² (D) y = 2x² + C
Answer
(A) y = x² + C — the general solution contains an arbitrary constant.
Determine the order of the differential equation d³y/dx³ + 2d²y/dx² + dy/dx = 0 by identifying its highest-order derivative.
Answer
Highest-order derivative: d³y/dx³ ∴ Order = 3
Determine the degree of the differential equation (d²y/dx²)³ + (dy/dx)² = x.
Answer
Highest-order derivative: d²y/dx² Power of d²y/dx²: 3 ∴ Degree = 3
For the differential equation dy/dx = 2x, find a solution function and write the corresponding curve. State the name given to the curve represented by a solution function.
Answer
The curve represented by a solution function y = φ(x) is called the solution curve (integral curve) of the differential equation. On integrating, y = ∫2x dx = x² + C ∴ The solution function is y = x² + C, and its corresponding curve is the solution curve (integral curve) y = x² + C.
Find the integrating factor of dy/dx+(2x+1/x)y=x², for x>0.
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Verify that y = e^(−3x) satisfies d²y/dx² + 6dy/dx + 9y = 0. Find the first and second derivatives, substitute them into the equation, and check that its two sides are equal.
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For each differential equation below, determine its order and degree, if defined. Show how the highest-order derivative and its power determine the degree, and indicate whether each equation is polynomial in its derivatives: (i) (d²y/dx²)² + dy/dx = 0; (ii) sin(dy/dx) + d²y/dx² = 0; (iii) (d³y/dx³) + (d²y/dx²)² = 0. State the possible kind of values of order and defined degree.
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Solve the homogeneous differential equation dy/dx=1+y/x by using the substitution y=vx. Show the differentiation, separation of variables and integration, and express the general solution in terms of x and y.
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Find the degree of (d²y/dx²)⁴ + dy/dx = 0.
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44 more probable questions in this chapter, every one with a full answer.
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