Ordinary Differential Equations + Integral Calculus

ORDER;

The order of the highest order derivative present in the differential equation. Ex: d³y/dx³ + (dy/dx)⁴ = 0. Order is 3

DEGREE;

The highest power of the highest order derivative, after making the D.E. a polynomial in derivatives. Ex: d³y/dx³ + (dy/dx)⁴ = 0. Degree is 1

DEGREE NOT DEFINED;

Degree is not defined if the D.E. cannot be expressed as a polynomial in its derivatives. Ex: sin(dy/dx) + y = 0. Look for derivatives inside transcendental functions like sin, cos, log, e^x.

FORMATION PROCESS (DIFFERENTIAL EQUATIONS);

1. Write the equation for the family of curves. 2. Count the number of arbitrary constants, 'n'. 3. Differentiate the equation 'n' times. 4. Eliminate all 'n' constants using the 'n+1' equations. The order of the resulting D.E. will be equal to 'n'.

EXAMPLE: FAMILY Y = MX;

One constant 'm'. Differentiate once. y = mx … (1), dy/dx = m … (2). Substitute (2) in (1): y = (dy/dx)x or x dy - y dx = 0

STANDARD FORM (VARIABLE SEPARABLE FORM);

f(y) dy = g(x) dx

SOLUTION METHOD (VARIABLE SEPARABLE FORM);

Integrate both sides: ∫ f(y) dy = ∫ g(x) dx + C. Goal: Get all 'y' terms with dy and all 'x' terms with dx.

REDUCIBLE FORM (VARIABLE SEPARABLE FORM);

dy/dx = f(ax + by + c). Substitution: Let z = ax + by + c. Then solve using variable separation for z and x.

IDENTIFICATION (HOMOGENEOUS EQUATIONS);

A D.E. dy/dx = F(x, y) is homogeneous if F(x, y) is a homogeneous function of degree zero. Check: F(λx, λy) = F(x, y) Or, dy/dx = g(y/x).

SOLUTION METHOD (Y/X FORM HOMOGENEOUS);

If dy/dx = g(y/x): 1. Substitute y = vx. 2. Then dy/dx = v + x(dv/dx). 3. Equation becomes variable separable in 'v' and 'x'. 4. Solve and substitute back v = y/x.

SOLUTION METHOD (X/Y FORM HOMOGENEOUS);

If dx/dy = h(x/y): 1. Substitute x = vy. 2. Then dx/dy = v + y(dv/dy). 3. Equation becomes variable separable in 'v' and 'y'. 4. Solve and substitute back v = x/y.

STANDARD FORM (LDE IN Y);

dy/dx + P(x)y = Q(x) where P and Q are functions of x only.

INTEGRATING FACTOR (I.F. FOR LDE IN Y);

I.F. = e^(∫ P(x) dx). This factor makes the LHS a perfect derivative.

GENERAL SOLUTION (LDE IN Y);

y × (I.F.) = ∫ [Q(x) × (I.F.)] dx + C

LDE IN X (DX/DY FORM);

dx/dy + P(y)x = Q(y). I.F. = e^(∫ P(y) dy). Solution: x × (I.F.) = ∫ [Q(y) × (I.F.)] dy + C

OPTION VERIFICATION (BITSAT SHORTCUT);

For 'find the solution' problems, it can be faster to differentiate the options and check which one satisfies the given D.E. and initial conditions. Best for complex D.E.s where integration is long.

HOMOGENEOUS CHECK TRICK;

Quickly check if all terms have the same total degree in x and y. Ex: (x² + y²)dx - 2xy dy = 0. Degrees are 2, 2, 2. So it's homogeneous.

BITSAT EXAM TACTICS (D.E.);

1. First, classify the D.E.: Is it Separable? Homogeneous? Linear? This is the most crucial step. 2. Don't forget the constant of integration '+C'. It's a common trap in MCQ options. 3. For 'formation' problems, count the arbitrary constants. That number is the order of the D.E. you need to form. 4. If a D.E. is both homogeneous and linear (e.g., dy/dx + y/x = 0), the linear method is usually faster. 5. Pay close attention to dy/dx vs dx/dy. The form of the LDE can be inverted.

VARIABLE SEPARABLE IDENTIFICATION;

How to Identify: Can group all y, dy terms on one side and all x, dx terms on the other. f(y)dy = g(x)dx. Key Step: Direct Integration.

HOMOGENEOUS IDENTIFICATION;

How to Identify: All terms have the same degree. Can be written as dy/dx = F(y/x). Key Step: Substitute y = vx.

LINEAR (LDE) IDENTIFICATION;

How to Identify: Standard form: dy/dx + P(x)y = Q(x). 'y' and 'dy/dx' are of degree 1. Key Step: Find Integrating Factor (I.F.)

POWER RULE INTEGRAL;

I.F. = ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ -1)

LOGARITHMIC RULE INTEGRAL;

∫ (1/x) dx = ln|x| + C

EXPONENTIAL RULES INTEGRALS;

∫ e^x dx = e^x + C | ∫ a^x dx = a^x / ln(a) + C

BASIC TRIGONOMETRIC INTEGRALS;

∫ sin(x) dx = -cos(x) + C | ∫ cos(x) dx = sin(x) + C

SECANT/COSECANT SQUARED INTEGRALS;

∫ sec²(x) dx = tan(x) + C | ∫ csc²(x) dx = -cot(x) + C

OTHER TRIG PRODUCTS INTEGRALS;

∫ sec(x)tan(x) dx = sec(x) + C | ∫ csc(x)cot(x) dx = -csc(x) + C

INVERSE TAN FORM INTEGRAL;

∫ dx / (x² + a²) = (1/a) tan⁻¹(x/a) + C

INVERSE SINE FORM INTEGRAL;

∫ dx / √(a² - x²) = sin⁻¹(x/a) + C

LOGARITHMIC FORMS INTEGRAL;

∫ dx / (x² - a²) = (1/2a) ln|(x-a)/(x+a)| + C

OTHER LOG FORMS INTEGRAL;

∫ dx / √(x² ± a²) = ln|x + √(x² ± a²)| + C

CORE TRIG INTEGRALS;

∫ tan(x) dx = ln|sec(x)| + C | ∫ cot(x) dx = ln|sin(x)| + C

SECANT / COSECANT INTEGRALS;

∫ sec(x) dx = ln|sec(x)+tan(x)| | ∫ csc(x) dx = ln|csc(x)-cot(x)|

INTEGRATION BY PARTS (ILATE);

∫ u v dx = u ∫v dx - ∫(u' ∫v dx) dx. Choose 'u' using ILATE priority: Inverse Log, Algebraic, Trig, Exponential. 'v' is the other part.

INTEGRATION BY SUBSTITUTION;

∫ f(g(x)) g'(x) dx. Let t = g(x), then dt = g'(x)dx. Becomes: ∫ f(t) dt. Look for a function and its derivative pair.

PARTIAL FRACTIONS (LINEAR);

For P(x) / (x-a)(x-b): Decompose to A/(x-a) + B/(x-b) and integrate term-by-term.

SPECIAL FORM: E^x [F(X) + F'(X)];

∫ e^x [f(x) + f'(x)] dx = e^x f(x) + C. A very common BITSAT shortcut. Identify f(x) and its derivative f'(x) quickly.

FUNDAMENTAL THEOREM (FTC);

∫ₐᵇ f(x) dx = F(b) - F(a) where F'(x) = f(x).

KING'S PROPERTY (MOST IMPORTANT);

∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx. Use this to simplify complex integrands, especially in trigonometry.

EVEN-ODD FUNCTION PROPERTY;

∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx, if f(x) is even | 0, if f(x) is odd

QUEEN'S PROPERTY;

∫₀²ᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx, if f(2a-x) = f(x) | 0, if f(2a-x) = -f(x)

BITSAT INTEGRATION STRATEGY;

1. King's Property (∫f(a+b-x)) solves over 50% of tough definite integral problems. Always try it first. 2. For ∫e^x(f(x)+f'(x))dx, quickly identify f(x) and f'(x). Common pairs: (ln x, 1/x), (tan⁻¹x, 1/(1+x²)). 3. See limits like -a to a? Immediately check if the function is even or odd. It's a massive time-saver. 4. Don't waste time on complex partial fractions. BITSAT questions use simple linear or repeated factors. 5. Memorize the results for ∫√(a²-x²), ∫√(x²+a²), etc. They are often asked directly.

AREA WITH X-AXIS;

Area = ∫ₐᵇ |y| dx = ∫ₐᵇ |f(x)| dx. Use absolute value. If f(x) is below the x-axis, the integral is negative, so area is -∫f(x)dx.

AREA WITH Y-AXIS;

Area = ∫_c^d |x| dy = ∫_c^d |g(y)| dy. Use when the curve is defined as x in terms of y.

AREA BETWEEN TWO CURVES;

Area = ∫ₐᵇ |f(x) - g(x)| dx. Integrate (Top Curve - Bottom Curve) over the interval defined by their intersection points.

INDEFINITE INTEGRAL VS DEFINITE INTEGRAL (RESULT);

Indefinite: A family of functions (f(x) + C) | Definite: A single numerical value

INDEFINITE INTEGRAL VS DEFINITE INTEGRAL (CONSTANT 'C');

Indefinite: Always present | Definite: Not present (cancels out)

INDEFINITE INTEGRAL VS DEFINITE INTEGRAL (LIMITS);

Indefinite: Has no limits of integration | Definite: Has upper and lower limits

INDEFINITE INTEGRAL VS DEFINITE INTEGRAL (REPRESENTS);

Indefinite: Antiderivative of a function | Definite: Net signed area under the curve