Question

If \[ \frac{a}{a_1}=\frac{b}{b_1}, \] then the substitution to be used to solve the differential equation \[ \frac{dy}{dx}=\frac{ax+by+c}{a_1x+b_1y+c_1} \] by using separation of variables is

• A. \(x=X+h,\; y=Y+k\)
• B. \(ax+by=Z\)
• C. \(y=V(x)\cdot x\)
• D. \(x=at,\; y=bt\)

Hint:

Whenever \[ \frac{a}{a_1}=\frac{b}{b_1}, \] the linear expressions in the numerator and denominator are proportional. In such cases, use a substitution involving that common linear combination, such as \(Z=ax+by\) or \(Z=a_1x+b_1y\).

Answer 1

The Correct Option is B

Step 1: Recognize the condition a/a1=b/b1.
When \(\frac{a}{a_1}=\frac{b}{b_1}=k\) (say), the lines \(ax+by\) and \(a_1x+b_1y\) are parallel (both equal \(k(a_1x+b_1y)\)). So the RHS has the form \(\frac{k(a_1x+b_1y)+c}{a_1x+b_1y+c_1}\), which is a function of \(a_1x+b_1y\) (or equivalently \(ax+by\)).

Step 2: Choose the correct substitution.
Setting \(Z=ax+by\) converts the equation into a separable ODE in \(Z\) and \(x\): \(\frac{dZ}{dx}=a+b\frac{dy}{dx}\). Hence the substitution is \[\boxed{ax+by=Z}\]