Question and Answers Forum

All Questions      Topic List

Differential Equation Questions

Previous in All Question      Next in All Question      

Previous in Differential Equation      Next in Differential Equation      

Question Number 120058 by bramlexs22 last updated on 29/Oct/20

 (d^2 y/dx) +x (dy/dx) −y=0

$$\:\frac{{d}^{\mathrm{2}} {y}}{{dx}}\:+{x}\:\frac{{dy}}{{dx}}\:−{y}=\mathrm{0} \\ $$

Answered by Olaf last updated on 29/Oct/20

  y′′+xy′−y = 0 (1)  y = xu  y′ = xu′+u  y′′ = xu′′+2u′    (1) : xu′′+2u′+x(xu′+u)−xu = 0  xu′′+2u′+x^2 u′ = 0  ((u′′)/(u′)) = −((2+x^2 )/x) = −(2/x)−x  ln∣u′∣ = −2ln∣x∣−(x^2 /2)+C_1   u′ = C_2 e^(ln(1/x^2 )) e^(−(x^2 /2))  = (C_2 /x^2 )e^(−(x^2 /2))   u = −C_2 [(e^(−(x^2 /2)) /x)+(√(π/2))erf((x/( (√2))))]+C_3   (By parts)    y = ux = −C_2 [e^(−(x^2 /2)) +(√(π/2))xerf((x/( (√2))))]+C_3 x

$$ \\ $$$${y}''+{xy}'−{y}\:=\:\mathrm{0}\:\left(\mathrm{1}\right) \\ $$$${y}\:=\:{xu} \\ $$$${y}'\:=\:{xu}'+{u} \\ $$$${y}''\:=\:{xu}''+\mathrm{2}{u}' \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\::\:{xu}''+\mathrm{2}{u}'+{x}\left({xu}'+{u}\right)−{xu}\:=\:\mathrm{0} \\ $$$${xu}''+\mathrm{2}{u}'+{x}^{\mathrm{2}} {u}'\:=\:\mathrm{0} \\ $$$$\frac{{u}''}{{u}'}\:=\:−\frac{\mathrm{2}+{x}^{\mathrm{2}} }{{x}}\:=\:−\frac{\mathrm{2}}{{x}}−{x} \\ $$$$\mathrm{ln}\mid{u}'\mid\:=\:−\mathrm{2ln}\mid{x}\mid−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\mathrm{C}_{\mathrm{1}} \\ $$$${u}'\:=\:\mathrm{C}_{\mathrm{2}} {e}^{\mathrm{ln}\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} {e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \:=\:\frac{\mathrm{C}_{\mathrm{2}} }{{x}^{\mathrm{2}} }{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \\ $$$${u}\:=\:−\mathrm{C}_{\mathrm{2}} \left[\frac{{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} }{{x}}+\sqrt{\frac{\pi}{\mathrm{2}}}\mathrm{erf}\left(\frac{{x}}{\:\sqrt{\mathrm{2}}}\right)\right]+\mathrm{C}_{\mathrm{3}} \\ $$$$\left(\mathrm{By}\:\mathrm{parts}\right) \\ $$$$ \\ $$$${y}\:=\:{ux}\:=\:−\mathrm{C}_{\mathrm{2}} \left[{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} +\sqrt{\frac{\pi}{\mathrm{2}}}{x}\mathrm{erf}\left(\frac{{x}}{\:\sqrt{\mathrm{2}}}\right)\right]+\mathrm{C}_{\mathrm{3}} {x} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com