Answer to Question #295579 in Differential Equations for chinchu

Question

show that y= c1e^x +c2^2x is the general soloution of ydobledash -3ydash+2y=0

Answer 1

Solution:

$\begin{aligned} & y=c_{1} e^{x}+c_{2} e^{2 x} \\ & \text { To verify the equation } \\ & y^{\prime \prime}-3 y^{\prime}+2 y=0 \\ y^{\prime}=\frac{d y}{d x}=c_{1} e^{x}+2 c_{2} e^{2 x} \\ y^{\prime \prime}=\frac{d^{2} y}{d x^{2}}=c_{1} e^{x}+4 c_{2} e^{2 x} \\ \Rightarrow & c_{1} e^{x}+4 c_{2} e^{2 x}-3\left(c_{1} e^{x}+2 c_{2} e^{2 x}\right) \\ &+2\left(c_{1} e^{x}+c_{2} e^{2 x}\right) \\ \Rightarrow & c_{1} e^{x}+4 c_{2} e^{2 x}-3 c_{1} e^{\prime x}-6 c_{2} e^{2 x} \\ &+2 c_{1} e^{x}+2 c_{2} e^{2 x} \\ \Rightarrow & 6 c_{2} e^{2 x}-6 c_{2} e^{2 x} \\ \Rightarrow & 0 \end{aligned}$