NDA Differential Equations: Order, Degree and First-Order Solutions

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Definition of order and degree; formation of differential equations by examples; general and particular solutions; first-order and first-degree differential equations of various types; applications to growth and decay.

Definition of order and degree; formation of differential equations by examples; general and particular solutions; first-order and first-degree differential equations of various types; applications to growth and decay.

Differentiation, integration, algebraic rearrangement and logarithms.

• Differential equation
• Order
• Degree, where polynomial in derivatives
• General solution
• Particular solution

• Separable form: dy/dx = g(x)h(y) -> dy/h(y)=g(x)dx, when division is valid
• Linear first order: dy/dx + P(x)y = Q(x), integrating factor IF=e^(int P dx)
• After IF: y*IF = int Q*IF dx + C
• Growth / decay model: dy/dt=ky -> y=Ce^(kt)

• Degree is defined after the equation is made polynomial in derivatives; it may be undefined otherwise.
• When separating variables, do not divide by an expression that may be zero without checking lost constant solutions.
• Linear integrating-factor method requires standard first-order linear form.
• Growth / decay sign and initial condition determine k and C.

1. dy/dx=3x^2 gives y=x^3+C. If y(0)=2, particular solution y=x^3+2.
2. dy/dx + y = e^x: IF=e^x, so d(y e^x)/dx=e^(2x); y e^x=(1/2)e^(2x)+C, hence y=(1/2)e^x+Ce^(-x).
3. If quantity doubles from Q0 at t=0 to 2Q0 at t=T in y=Q0 e^(kt), then k=ln2/T.

• Confusing order with degree.
• Forgetting constant of integration.
• Dropping a solution by dividing by y or another factor.
• Using an integrating factor for a non-linear equation without first checking form.
• Sign error in exponential decay.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Identify the equation type before manipulating it. If separable, separate cleanly and inspect possible lost solutions. If first-order linear, write P and Q explicitly before calculating the integrating factor. Apply initial conditions only after obtaining the general solution unless the method clearly benefits from earlier substitution.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.