(a) Find the general solution of the following differential equation: $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ OR (b) Find the particular solution of the differential equation $xy \frac{dy}{dx} = (x + 2)(y + 2)$ , given that $y(1) = -1$ .

Q: (a) Find the general solution of the following differential equation: $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ OR (b) Find the particular solution of the differential equation $xy \frac{dy}{dx} = (x + 2)(y + 2)$, given that $y(1) = -1$.

**Option (a):** Dividing by $x^2$ gives $\frac{dy}{dx} = 1 + y/x + (y/x)^2$. Substituting $y=vx$ reduces this to the separable form $\frac{dv}{1+v^2} = \frac{dx}{x}$, integrating to $\tan^{-1}v = \ln|x| + C$. The general solution is **$\tan^{-1}(y/x) = \ln|x| + C$**. **Option (b):** Separating variables gives $\left[1 - \frac{2}{y+2}\right]dy = \left[1 + \frac{2}{x}\right]dx$. Integrating: $y - 2\ln|y+2| = x + 2\ln|x| + C$. Using $y(1) = -1$: $-1 - 0 = 1 + 0 + C \implies C = -2$. The particular solution is **$y - x + 2 = 2\ln|x(y+2)|$**.

Differential Equations

Differential Equations — Homogeneous and Variable Separable Forms

Official Answer

Option (a): Dividing by $x^2$ gives $\frac{dy}{dx} = 1 + y/x + (y/x)^2$ . Substituting $y=vx$ reduces this to the separable form $\frac{dv}{1+v^2} = \frac{dx}{x}$ , integrating to $\tan^{-1}v = \ln|x| + C$ . The general solution is $\tan^{-1}(y/x) = \ln|x| + C$ .

Option (b): Separating variables gives $\left[1 - \frac{2}{y+2}\right]dy = \left[1 + \frac{2}{x}\right]dx$ . Integrating: $y - 2\ln|y+2| = x + 2\ln|x| + C$ . Using $y(1) = -1$ : $-1 - 0 = 1 + 0 + C \implies C = -2$ . The particular solution is $y - x + 2 = 2\ln|x(y+2)|$ .

homogeneous differential equationvariable separablegeneral solutionparticular solutioninitial conditionsubstitution y=vx

Marking Scheme

11 mark: correct identification of equation type and substitution/separation setup ( $y=vx$ for a; splitting fractions for b).
21 mark: correct integration to reach $\tan^{-1}v = \ln|x| + C$ (a) or $y - 2\ln|y+2| = x + 2\ln|x| + C$ (b).
31 mark: correct final general solution in terms of $y/x$ (a), or correct evaluation of C using the initial condition and final particular solution (b).

Hint

For (a), divide by $x^2$ and substitute $y=vx$ (homogeneous equation). For (b), separate variables using $\frac{y}{y+2} = 1 - \frac{2}{y+2}$ and $\frac{x+2}{x} = 1 + \frac{2}{x}$ .

Quick Oral Answer

Part (a) uses the substitution $y=vx$ to get $\tan^{-1}(y/x) = \ln|x| + C$ ; part (b) separates variables to get $y - 2\ln|y+2| = x + 2\ln|x| + C$ , and using $y(1) = -1$ gives $C = -2$ .

Analysis & Explanation

This OR question tests the two most common first-order differential equation techniques: the homogeneous substitution method and the variable separable method.

Concept: Part (a)'s equation is homogeneous because every term $(x^2, xy, y^2)$ has the same total degree; dividing by $x^2$ expresses it purely in terms of $v=y/x$ , converting it into a separable equation. Part (b) is directly separable once each side is split into a constant plus a proper fraction ( $1 - \frac{2}{y+2}$ and $1 + \frac{2}{x}$ ), since $\frac{y}{y+2}$ and $\frac{x+2}{x}$ are not directly integrable as written.

Exam trap: In part (a), a frequent error is forgetting the product rule when substituting $\frac{dy}{dx} = v + x\frac{dv}{dx}$ . In part (b), students often try to integrate $\frac{y}{y+2}$ directly without splitting it first, giving an incorrect log form, or apply the initial condition before fully simplifying.

Real-world link: Homogeneous and variable-separable differential equations model population growth, radioactive decay, and rate-based mixing problems where the rate of change depends proportionally on the current state.

Common Mistakes

1Forgetting the product rule term $x(dv/dx)$ when substituting $y=vx$ in part (a).
2Integrating $\frac{y}{y+2}$ or $\frac{x+2}{x}$ directly without splitting into a constant plus a proper fraction in part (b), leading to an incorrect logarithmic term.
3Applying the initial condition $y(1) = -1$ before simplifying the general solution fully, causing an incorrect value of the constant C.

Interesting Facts

Homogeneous differential equations of this type were systematically studied by Johann Bernoulli in the late 17th century as part of the early development of calculus-based modelling.

Variable-separable equations of the form in part (b) are the mathematical basis of the logistic growth model used in population biology and epidemiology.

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Frequently Asked Questions

How do you recognise a homogeneous differential equation?

A differential equation $\frac{dy}{dx} = f(x,y)$ is homogeneous if f(x,y) can be written as a function of $y/x$ alone, i.e. every term has the same total degree in x and y, as with $x^2, xy, y^2$ in part (a).

Why split $\frac{y}{y+2}$ into $1 - \frac{2}{y+2}$ in part (b)?

Direct integration of $\frac{y}{y+2}$ is not a standard form; rewriting it as 1 minus a proper fraction converts it into two integrable standard terms, $y$ and $\ln|y+2|$ .