ilovepdf_merged (6).pdf). Content covers sections 32.1–32.x.A differential equation involves a function and its derivatives. Order = highest derivative; degree = power of that derivative when the equation is polynomial in derivatives.
Write as g(y) dy = f(x) dx and integrate both sides. Include +C.
Homogeneous: dy/dx = F(y/x); substitute v=y/x. Linear: dy/dx + P(x)y = Q(x); integrating factor e^(∫P dx).
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Differential Equations.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: cos x
Memorise and apply: sin x
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
4 question(s) · Sources: Board-style IVP, Board-style linear, Board-style order/degree, Board-style separable
PYQ1. State the order and degree of (y″)³ + y′ = 0.
Model Answer
1) Order 2, degree 3
Explanation
1) Highest derivative y″; raised to power 3 in polynomial form.
PYQ2. Solve dy/dx = y (general solution).
Model Answer
1) y = C eˣ
Explanation
1) dy/y = dx ⇒ ln|y|=x+C₁ ⇒ y=Ceˣ.
PYQ3. Solve dy/dx + y = eˣ.
Model Answer
1) y = (1/2)eˣ + C e^(−x)
Explanation
1) IF=eˣ; d/dx(yeˣ)=e^(2x); yeˣ=(1/2)e^(2x)+C; y=(1/2)eˣ+Ce^(−x).
PYQ4. Solve dy/dx=2x with y(0)=3.
Model Answer
1) y = x² + 3
Explanation
1) y=x²+C; y(0)=3 ⇒ C=3.
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
State the order and degree of (d²y/dx²)³ + dy/dx = 0.
1) Highest derivative is d²y/dx² ⇒ order 2.
2) It appears to power 3 and equation is polynomial in derivatives ⇒ degree 3.
Answer: Order 2, degree 3
Order from highest derivative; degree from its power when polynomial.
Second derivative cubed → order 2 degree 3.
If fractional powers of y″, degree may be undefined after clearing.
Do not call order 3.
Solve dy/dx = y (y>0).
1) dy/y = dx.
2) ln y = x + C₁ ⇒ y = e^(x+C₁)=A eˣ with A>0.
3) General often written y=Ceˣ (C≠0); including C=0 covers y=0 solution.
Answer: y = C eˣ
Variable separable; exponential solutions.
dy/y integrates to ln; exp both sides.
C arbitrary real if y=0 included.
Do not leave ln y = x without solving for y unless asked.
Solve y dy = x dx.
1) ∫ y dy = ∫ x dx ⇒ y²/2 = x²/2 + C₁ ⇒ y² − x² = C.
Answer: y² − x² = C
Separable equation integrates to a family of hyperbolas (difference of squares).
Integrate both sides, multiply by 2.
C = 2C₁.
Can write x²−y²=K by sign flip of constant.
Solve dy/dx + y = eˣ.
1) P=1 ⇒ IF=e^{∫dx}=eˣ.
2) d/dx(y eˣ)=eˣ eˣ=e^(2x).
3) y eˣ = (1/2)e^(2x)+C ⇒ y = (1/2)eˣ + C e^(−x).
Answer: y = ½ eˣ + C e^(−x)
Standard linear first-order method with integrating factor eˣ.
Multiply by eˣ, left side becomes product derivative.
Divide by IF at the end.
∫ e^(2x) dx = e^(2x)/2.
Verify y=x² is a solution of x y′ = 2y (x≠0).
1) y′=2x; x·2x=2x² and 2y=2x².
2) Equal ⇒ solution.
Answer: Verified: both sides equal 2x²
Direct substitution of y and y′ into the DE.
Derivative 2x; multiply by x matches 2y.
Verification does not find C; it checks a candidate.
Compute y′ carefully.
Solve dy/dx=2x with y(0)=3.
1) y=x²+C; 3=0+C ⇒ C=3.
2) y=x²+3.
Answer: y = x² + 3
General solution plus initial condition fixes C.
Integrate 2x get x²+C; use y(0)=3.
Particular solution has no free C.
y(0) means x=0.
State the order and degree of (dy/dx)² + y = x.
1) Highest derivative is dy/dx ⇒ order 1.
2) It appears to power 2 ⇒ degree 2.
Answer: Order 1, degree 2
Order is the highest derivative present; degree is its power in polynomial form.
First derivative squared.
y and x do not change order.
Check each algebraic step carefully.
Solve dy/dx = 3y (y>0).
1) dy/y = 3 dx.
2) ln y = 3x + C₁ ⇒ y = A e^{3x} (A>0).
3) General real form often written y = C e^{3x}.
Answer: y = C e^{3x}
Separable first-order equation.
Grow exponentially with rate 3.
C absorbs the sign of A.
Check each algebraic step carefully.
Solve dy/dx + y = 0.
1) P=1, IF=eˣ.
2) d/dx(y eˣ)=0 ⇒ y eˣ = C ⇒ y = C e^{−x}.
Answer: y = C e^{−x}
First-order linear with Q=0.
Exponential decay solutions.
IF multiplies both sides.
Check each algebraic step carefully.
Solve dy/dx = 2x with y(0)=4.
1) y = x² + C.
2) y(0)=4 ⇒ C=4.
3) y = x² + 4.
Answer: y = x² + 4
General solution plus initial condition.
Antiderivative of 2x is x².
Plug x=0 to fix C.
Check each algebraic step carefully.