Question

The degree of the differential equation is

• A. 3
• B. 1
• C. 2
• D. None of these

Answer 1

The correct option is B 1

As $m_1$, $m_2$, $m_3$ are roots of $m^3-9m^2+23m-15=0$
$\Rightarrow$ $(m-1)(m-3)(m-5)=0$
$\Rightarrow$ $m_1=1$, $m_2=3$, $m_3=5$
$\therefore$ Required solution is $y=c_1{e}^x+c_2{e}^{3x}+c_3{e}^{5x}$ (i)
On Differentiating w.r to x both sides we get
$\Rightarrow$ $y^{\prime }=c_1{e}^x+3c_2{e}^{3x}+5c_3{e}^{5x}$ (ii)
(using (ii) -(i))
$\Rightarrow$ $y^{\prime }-y=2c_2{e}^{3x}+4c_3{e}^{5x}$ (iii)
Again differentiating w.r to x both sides
$\Rightarrow$ $y^{\prime \prime }-y^{\prime }=6c_2{e}^{3x}+20c_3{e}^{5x}$ (iv)
using (iv)-3(iii) we get
$y^{\prime \prime }-4y^{\prime }+3y=8c_3{e}^{5x}$ (v)
$\Rightarrow$ $y^{\prime \prime \prime }-4y^{\prime \prime }+3y^{\prime }=8\times 5c_3{e}^{5x}$ (vi)
using (vi)-5(v) we get
$y^{\prime \prime \prime }-9y^{\prime \prime }+23y^{\prime }-15y=0$
or $\frac{d^{3}y}{d{x}^3}-9\frac{d^{2}y}{d{x}^2}+23\frac{dy}{dx}-15y=0$
Hence degree of differential equation is 1.