Differential Equations
Differential Equations Reducible to Homogeneous Form
Grade 12
Question:
<p>The solution of the differential equation $\frac{dy}{dx} = \frac{y - x + 1}{y + x + 5}$ is</p>
<p>(a) $\log [(y + 3)^2 + (x + 2)^2] + 2\tan^{-1}\left\{\frac{y + 3}{x + 2}\right\} = C$</p>
<p>(b) $\log [(y + 3)^2 + (x + 2)^2] + \tan^{-1}\left\{\frac{y + 3}{x + 2}\right\} = C$</p>
<p>(c) $\log [(y + 3)^2 - (x + 2)^2] + 2\cot^{-1}\left\{\frac{y + 3}{x + 2}\right\} = C$</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Differential equations of the form $\frac{dy}{dx} = \frac{ax + by + c}{dx + ey + f}$ where coefficients are not proportional can be reduced to homogeneous form using the substitution $x = X + h$ and $y = Y + k$.
<p><strong>Solution:</strong> The coefficients of $x$ and $y$ in the numerator and denominator of the expression for $\frac{dy}{dx}$ are not proportional. Such equations can be reduced to homogeneous form by taking new variables $X$ and $Y$ such that $x = X + h$ and $y = Y + k$, where $h$ and $k$ are constants to be determined.</p>
Correct Answer: A