Differential Equations
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Direct answer
The order of a differential equation is the order of its highest derivative, and its degree is the power of that highest derivative once the equation is made polynomial in derivatives. First-order equations in JEE reduce to three solvable types: variable separable (separate and integrate), homogeneous of the form dy/dx = F(y/x) (substitute y = vx), and linear dy/dx + P(x) y = Q(x) (multiply by the integrating factor e^integral of P dx).
What you must remember
- Order and degree: order = highest derivative present; degree = its power after clearing radicals and fractions in the derivatives — sqrt(1 + (dy/dx)^2) must be squared out before any degree is quoted.
- Variable separable: rewrite as f(y) dy = g(x) dx and integrate both sides; fix the constant of integration from the given boundary condition.
- Homogeneous type: dy/dx = F(y/x) yields to y = vx, so dy/dx = v + x dv/dx; separate in v and x, integrate, then substitute back v = y/x.
- Linear in y: dy/dx + P(x) y = Q(x) has integrating factor IF = e^integral P dx, and the solution is y × IF = integral of (Q × IF) dx + C.
- Linear in x: equations of the form dx/dy + P(y) x = Q(y) work identically — recognise which variable the linearity is in.
- Reducible equations: dy/dx = (ax + by + c)/(a'x + b'y + c') with the line pairs intersecting shifts the origin to the intersection point and becomes homogeneous.
- Forming equations: a family of curves with n arbitrary constants generates a differential equation of order n — differentiate n times and eliminate the constants.
Common confusion
Degree is the most misused word in this chapter. An equation containing sqrt(1 + (dy/dx)^2) has no degree until the radical is cleared; after squaring, the degree is 2 while the order is still 1. Students also misclassify homogeneity: dy/dx = (y/x) + 1 is homogeneous because the right side is a function of y/x, while dy/dx = (x + y + 1)/(x - y) is not, and needs the origin shift. Finally, the integrating factor belongs to the standard form with coefficient 1 on dy/dx — divide through before reading off P(x).
Exam-focused takeaway
JEE Main asks order-degree identification, variable separable and linear equations solved to a numerical constant via the boundary condition — mechanical marks with careful bookkeeping. JEE Advanced prefers equations reduced by substitution, differential equations formed from curve families, and solutions interleaved with curves or areas. Whatever the wrapper, the sequence is fixed: put the equation in standard form, name its type, apply the type's substitution or factor, and only then integrate — with the boundary condition applied at the very end.
Frequently asked questions
What is the degree of a differential equation?
The power of the highest-order derivative after the equation is made polynomial in all derivatives; radicals and fractions containing derivatives must be cleared first.
When is a first-order equation homogeneous?
When the right side can be written purely as a function of y/x, making the substitution y = vx separate the variables.
What is an integrating factor?
For dy/dx + P(x) y = Q(x), the factor e^integral P dx; multiplying through makes the left side the derivative of y × IF, which integrates directly.
How many arbitrary constants does the general solution contain?
Exactly as many as the order of the equation; a first-order solution carries one constant, fixed by one boundary condition.
How is a differential equation formed from a family of curves?
Differentiate as many times as there are constants, then eliminate them between the original and derived equations.