Exercise 4.2 Class 12 Math Solutions - Unit 4 Differential Equations, Punjab Board

Exercise 4.2 of Class 12 Mathematics presents the fundamental analytical techniques for solving First-Order, First-Degree Ordinary Differential Equations under the Punjab Board PCTB / PECTAA curriculum. The first method introduced is the Separation of Variables, applicable when an ODE can be factored algebraically into the form (g(y)\,dy = f(x)\,dx), enabling immediate term-by-term integration to obtain the General Solution containing an arbitrary constant (c). When initial boundary conditions ((x_0, y_0)) are provided, students substitute values to isolate (c), yielding the unique Particular Solution.

The second major section deals with Homogeneous Differential Equations of the form (\frac{dy}{dx} = f(x, y)), where (f(x, y)) is a homogeneous function of degree zero. Students learn two practical tests to identify homogeneity: the Same-Degree Test (checking that every polynomial term in the numerator and denominator has identical total power) and the Variable Substitution Test ((f(\lambda x, \lambda y) = \lambda^0 f(x, y))). To solve homogeneous ODEs, students apply the substitution (y = vx), which yields (\frac{dy}{dx} = v + x\frac{dv}{dx}), transforming the equation into a separable ODE in terms of (v) and (x). After integrating, the substitution (v = \frac{y}{x}) is back-substituted to express the solution in terms of the original variables.

For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 4.2 is one of the highest-yield sources of Section C Long Questions and conceptual Short Questions. These step-by-step solutions provide clean variable separation, explicit differentiation of substitutions, and systematic initial value substitutions for every problem.