Solve x²d²y/dx²+5xdy/dx+4y=xlogx

Question

Solve x²d²y/dx²+5xdy/dx+4y=xlogx

Answer 1

x²\(\frac{d^2y}{d\mathrm x^2}\) + 5x\(\frac{dy}{d\mathrm x}\) + 4y = x log x ...........(1)

let log x = z ⇒ x = e^z

\(\Rightarrow \frac{1}{\mathrm x}\)dx = dz  (differentiate both sides)

\(\Rightarrow \frac{\mathrm x}{d\mathrm x}=\frac{1}{dz}\)

\(\Rightarrow \mathrm x \frac{dy}{d\mathrm x}=\frac{dy}{dz}\)  ............(2)

differentiate eqⁿ (2) w.r.t x, we get

\(\mathrm x\frac{d^2y}{d\mathrm x^2}+\frac{dy}{d\mathrm x}\) \(=\frac{d^2y}{dz^2}\) \(\frac{dz}{d\mathrm x}\)

\(\Rightarrow \mathrm x\frac{d^2y}{d\mathrm x^2}+\frac{dy}{d\mathrm x}\) \(=\frac{1}{\mathrm x}\frac{d^2y}{dz^2}\)  \(\left(\because \frac{dz}{d\mathrm x}=\frac{1}{\mathrm x}\right)\)

\(\Rightarrow \mathrm x^2\frac{d^2y}{d\mathrm x^2}+\mathrm x\frac{dy}{d\mathrm x}\) \(=\frac{d^2y}{dz^2}\)

\(\Rightarrow \mathrm x^2\frac{d^2y}{d\mathrm x^2}=\frac{d^2y}{dz^2}-\mathrm x\frac{dy}{d\mathrm x}\)

\(\Rightarrow \mathrm x^2\frac{d^2y}{d\mathrm x^2}=\frac{d^2y}{dz^2}-\frac{dy}{dz}\) .........(3)   (From equation (2))

From equations (1), (2) and (3), we get

\(\frac{d^2y}{dz^2}-\frac{dy}{dz}+5\frac{dy}{dz}+4y\) \(=ze^2\) ............(4)

Its auxiliary equation is

m² – m + 5m + 4 = 0

⇒ m² + 4m + 4 = 0

⇒ (m + 2)² = 0

⇒ m = –2, –2

Therefore, C.F = (C₁ + C₂^z)e^–2z

P.I. \(=\frac{1}{D^2+4D+4}ze^z\)    \((\because D=\frac{d}{dz})\)

\(=\frac{1}{(D+2)^2}ze^z\)

= e^z \(\frac{1}{((D+1)+2)^2}z\)     (\(\because\) \(\frac{1}{f(D)}e^{an}V=e^{a\mathrm x}\frac{1}{f(D+a)}V\) where V is function of x)

\(=e^z\frac{1}{(D+3)^2}z\)

\(=\frac{e^z}{9} \left(1+\frac{D}{3}\right)^{-2}z\)

\(=\frac{e^z}{9}\)\((1-\frac{2D}{3}+3\times \frac{D^2}{9}....)z\)

\(=\frac{e^z}{9}(z-\frac{2Dz}{3}+0....)\)

\(=\frac{e^z}{9}(z-\frac{2}{3})\)   \((\because Dz=\frac{d}{dz}z=1)\)

Therefore, complete solution of equation (4) is

y = C.F + P.I

= (c₁ + c₂z)e^–2z + \(\frac{e^z}{9}\left(z-\frac{2}{3}\right)\) ..........(5)

put z = log x & e^z = x in equation (5),

we get y = (c₁ + c₂ log x) \(\times\) \(\frac{1}{\mathrm x^2}\) + \(\frac{\mathrm x}{9}\left(\log \mathrm x-\frac{2}{3}\right)\)...........(6)

Equation (6) represents solution of given differential equation.