Differential Equations
Numeric — value computation
Grade Class 12
Question:
<p>\\(y''-4y'+4y=0\\), \\(y(0)=1\\), \\(y'(0)=2\\). Find \\(y(1)\\).</p>
<span>\(e^2\)</span>
<span>\(2e^2\)</span>
<span>\(3e^2\)</span>
<span>\(e^2+1\)</span>
Step-by-Step Solution
Key Concept: Repeated root m=2: y=(C_1+C_2x)e^{2x}.
<div class='solution'><p>Characteristic equation: $m^2-4m+4=(m-2)^2=0$. Repeated root $m=2$. $y=(C_1+C_2x)e^{2x}$. $y(0)=1$: $C_1=1$. $y'=(C_2+2C_1+2C_2x)e^{2x}$. $y'(0)=C_2+2C_1=2$: $C_2=0$. So $y=e^{2x}$. $y(1)=e^2$. <strong>Answer: (1)</strong> $e^2$.</p></div>
Correct Answer: 1