Question

The substitution required to reduce the differential equation \(t^2 dx + (x^2 - tx + t^2) dt = 0\) to a differential equation which can be solved by variables separable method is

• A. \(t = Vx\)
• B. \(ax + bt = Z\)
• C. \(V = tx^2\)
• D. \(x = tV^2\)

Hint:

For a homogeneous differential equation \(\frac{dy}{dx} = f(y/x)\), use \(y=vx\). For \(\frac{dx}{dy} = f(x/y)\), use \(x=vy\). Here, variables are \(x, t\), so \(x=vt\) or \(t=vx\) are the candidates.

Answer 1

The Correct Option is A

Step 1: Analyze the Equation:
Rewrite the differential equation: \[ t^2 dx = -(x^2 - tx + t^2) dt \] \[ \frac{dx}{dt} = -\frac{x^2 - tx + t^2}{t^2} = -\left( \left(\frac{x}{t}\right)^2 - \frac{x}{t} + 1 \right) \] The equation is a homogeneous differential equation because it can be expressed as a function of \(x/t\).
Step 2: Identify Substitution:
Standard substitutions for homogeneous equations are \(x = vt\) (which implies \(v = x/t\)) or \(t = vx\) (which implies \(v = t/x\)). The goal is to separate variables. Option (A) is \(t = Vx\). This is one of the valid substitutions for homogeneous equations (often used when \(dx/dt\) is a function of \(t/x\), but also valid here algebraically). Let's check if it works: If \(t = Vx\), then \(dt = V dx + x dV\). Substitute into the original form \(t^2 dx + (x^2 - tx + t^2) dt = 0\): Since it's homogeneous, this substitution will reduce it to a separable form involving \(V\) and \(x\). Specifically, substituting \(t=Vx\) (or equivalently \(x/t = 1/V\)) into \(\frac{dx}{dt} = f(x/t)\) is mathematically sound for separation. Thus, the substitution \(t = Vx\) is the correct choice among the options.