Differential Equations
Linear and Nonlinear ODEs
Grade 12
Question:
<p>Taking \(y\) as the dependent and \(x\) as the independent variable, which of the following ordinary differential equations are not linear?</p>
<p>(A) \(y'y'' + y^2 = x^2\)</p>
<p>(B) \(x^2y'' - xy' + 6y = \ln x\)</p>
<p>(C) \([1 + (y')^2]^{1/2} = 5y\)</p>
<p>(D) \(y' = \frac{x}{y}\)</p>
Step-by-Step Solution
Key Concept: A differential equation is linear if the dependent variable and all its derivatives appear with power 1 and there are no products between them.
<p>A linear differential equation in $y$ has the form: $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = f(x)$</p><p>The dependent variable and its derivatives must appear linearly (power 1 only).</p><p>(A) $y'y'' + y^2 = x^2$ — Contains $y' \cdot y''$ (product of derivatives) and $y^2$ (non-linear in $y$). <strong>NOT LINEAR</strong></p><p>(B) $x^2y'' - xy' + 6y = \ln x$ — Linear in $y, y', y''$. LINEAR</p><p>(C) $[1 + (y')^2]^{1/2} = 5y$ — Contains non-linear expression $\sqrt{1 + (y')^2}$. <strong>NOT LINEAR</strong></p><p>(D) $y' = \frac{x}{y}$ — This can be written as $yy' = x$, which is non-linear in $y$. <strong>NOT LINEAR</strong></p><p>All (A), (C), (D) are not linear.</p>
Correct Answer: A