Differential Equations

A differential equation (DE) is an equation involving derivatives of an unknown function. In JEE Main, the focus is on ordinary differential equations (ODEs) of first order and first degree, along with select higher-order types. The order of a DE is the highest order derivative present, and the degree is the power of the highest order derivative (when the equation is polynomial in derivatives).

Formation of Differential Equations: Given a family of curves with n arbitrary constants, differentiate n times and eliminate the constants to obtain the DE. The resulting DE has order n. Example: y = Ae^(2x) + Be^(-x) has 2 constants, so differentiate twice and eliminate A, B to get y'' - y' - 2y = 0.

Variable Separable Form: If the DE can be written as f(x) dx = g(y) dy, integrate both sides directly: integral of f(x) dx = integral of g(y) dy + C

Homogeneous Differential Equations: A DE dy/dx = f(x, y) is homogeneous if f(tx, ty) = f(x, y) for all t. Equivalently, dy/dx = g(y/x). Substitute y = vx where v = y/x, then dy/dx = v + x(dv/dx). The equation becomes separable in v and x.

Test for homogeneity: Each term in the numerator and denominator should have the same total degree in x and y.

Linear First-Order DE: Standard form: dy/dx + P(x)*y = Q(x). The integrating factor (IF) is e^(integral of P(x) dx). The solution is: y * IF = integral of (Q(x) * IF) dx + C

If the equation is dx/dy + P(y)*x = Q(y), treat x as the dependent variable with IF = e^(integral of P(y) dy).

Bernoulli's Equation: dy/dx + P(x)*y = Q(x)*y^n (n != 0, 1). Substitute v = y^(1-n) to convert to a linear equation: dv/dx + (1-n)*P(x)*v = (1-n)*Q(x).

Exact Differential Equations: M dx + N dy = 0 is exact if dM/dy = dN/dx. Then there exists F(x,y) such that dF = M dx + N dy = 0, so F(x,y) = C. In JEE, recognizing exact differentials like d(xy) = x dy + y dx, d(y/x) = (x dy - y dx)/$x^{2}$ is more useful than the general theory.

Common Exact Differentials to Recognize:

• x dy + y dx = d(xy)
• x dy - y dx = $x^{2}$ d(y/x) = -$y^{2}$ d(x/y)
• (x dy - y dx)/($x^{2}$ + $y^{2}$) = d(tan^(-1)(y/x))
• (x dx + y dy)/($x^{2}$ + $y^{2}$) = ($\frac{1}{2}$) d(ln($x^{2}$ + $y^{2}$))

Applications: Growth/decay (dy/dt = ky), Newton's cooling, orthogonal trajectories (replace dy/dx by -dx/dy in the family's DE).

The key problem-solving concept is identifying the TYPE of the differential equation (separable, homogeneous, linear, Bernoulli, or exact) and applying the corresponding standard method.