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
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
Show 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 \(Z=a_1x+b_1y\).
Updated On: Jun 26, 2026
- \(x=X+h,\; y=Y+k\)
- \(ax+by=Z\)
- \(y=V(x)\cdot x\)
- \(x=at,\; y=bt\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Analyze the given differential equation.
We are given \[ \frac{dy}{dx} = \frac{ax+by+c} {a_1x+b_1y+c_1} \] with the condition \[ \frac{a}{a_1} = \frac{b}{b_1}. \] Let \[ \frac{a}{a_1} = \frac{b}{b_1} = k. \] Then, \[ a=ka_1, \qquad b=kb_1. \]
Step 2: Rewrite the numerator.
Using the above relations, \[ ax+by+c = k(a_1x+b_1y)+c. \] Hence the differential equation becomes \[ \frac{dy}{dx} = \frac{k(a_1x+b_1y)+c} {a_1x+b_1y+c_1}. \] Observe that both numerator and denominator contain the same linear expression \[ a_1x+b_1y. \]
Step 3: Choose an appropriate substitution.
Let \[ Z=a_1x+b_1y. \] Since \(a\) and \(b\) are proportional to \(a_1\) and \(b_1\), an equivalent substitution is \[ Z=ax+by. \] Then, \[ \frac{dZ}{dx} = a+b\frac{dy}{dx}. \] Substituting the given differential equation into this expression converts the equation into a function involving only \(Z\), making it separable.
Step 4: Verify the method.
After substitution, the differential equation reduces to the form \[ \frac{dZ}{dx}=F(Z), \] which can be written as \[ \frac{dZ}{F(Z)}=dx. \] Thus, the variables become separable.
Step 5: Final conclusion.
Therefore, the required substitution is \[ \boxed{ax+by=Z}. \]
We are given \[ \frac{dy}{dx} = \frac{ax+by+c} {a_1x+b_1y+c_1} \] with the condition \[ \frac{a}{a_1} = \frac{b}{b_1}. \] Let \[ \frac{a}{a_1} = \frac{b}{b_1} = k. \] Then, \[ a=ka_1, \qquad b=kb_1. \]
Step 2: Rewrite the numerator.
Using the above relations, \[ ax+by+c = k(a_1x+b_1y)+c. \] Hence the differential equation becomes \[ \frac{dy}{dx} = \frac{k(a_1x+b_1y)+c} {a_1x+b_1y+c_1}. \] Observe that both numerator and denominator contain the same linear expression \[ a_1x+b_1y. \]
Step 3: Choose an appropriate substitution.
Let \[ Z=a_1x+b_1y. \] Since \(a\) and \(b\) are proportional to \(a_1\) and \(b_1\), an equivalent substitution is \[ Z=ax+by. \] Then, \[ \frac{dZ}{dx} = a+b\frac{dy}{dx}. \] Substituting the given differential equation into this expression converts the equation into a function involving only \(Z\), making it separable.
Step 4: Verify the method.
After substitution, the differential equation reduces to the form \[ \frac{dZ}{dx}=F(Z), \] which can be written as \[ \frac{dZ}{F(Z)}=dx. \] Thus, the variables become separable.
Step 5: Final conclusion.
Therefore, the required substitution is \[ \boxed{ax+by=Z}. \]
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