Question dy = syrx, Q7. Consider the following initial Value Problem y (o)=1 dx Use Euler's Method to estimate the solution on the interval [0, 1] with 5 steps. g(x) on

Ques: Gnsider the following Initial Value problem\frac{d y}{d x}=5 y+x \quad, y(0)=1Whe Euler's methoo to costimate the selution y(x) on the inferval [0,1] werth 5 stop.8 f^{n}=-\quad Given \quad \frac{d y}{d x}=5 y+x\begin{array}{l}y(0)=1 \\x=0 \Rightarrow y=1\end{array}\frac{d y}{d x}-5 y=xwe knoro that\frac{d y}{d x}+\mu y=\thetawher p, and \theta function of x and antontp=-5, \theta=xNow Intecgeting facter (I.F)\begin{aligned}I \cdot F & =e^{\int p d x} \\& =e^{\int-5 d x}=e^{-5 \int d x}=e^{-5 x} \\I \cdot F & =e^{-5 x}\end{aligned}Now Gereval sel"y(I, F)=\int \theta(I \cdot F) d xput vake of I.F and \thetaSoy\left(e^{-5 x}\right) \doteq \int \frac{x \cdot e^{-5 x}}{I} \cdot d xwe know thatAlgebra furction Nuenter I and expenitial function Nambor III\int I \cdot I d x=I \int \pi d x-\int\left(\frac{d}{d x}(I) \int d x\right) d xNow y\left(e^{-5 x}\right)=x \int e^{-5 x} d x-\left(\frac{d x}{d x}(x) \int e^{-5 x} d x\right) d xWe know that\begin{array}{l}\int e^{a x} d x=\frac{1}{a} e^{a x}+c \quad, \frac{d}{d x}(x)=1 \\\Rightarrow y\left(e^{-5 x}\right)=\frac{x e^{-5 x}}{-5}-\int \cdot \frac{e^{-5 x}}{-5} d x+c \\\Rightarrow y\left(e^{-5 x}\right)=-\frac{1 x}{5} e^{-5 x}+\frac{1}{5}\left(\frac{e^{-5 x}}{5}\right)+c \\\Rightarrow y\left(e^{-5 x}\right)=-\frac{1 x}{5} e^{-5 x}+\left(\frac{-1}{25}\right) e^{-5 x}+c \\\Rightarrow y e^{-5 x}=e^{-5 x}\left(-\frac{1 x}{5}-\frac{1}{25}\right)+c \\\Rightarrow \quad y=\frac{-x}{5}+\frac{1}{25}+c \\\end{array}Now given y(0)=1x=0 \Rightarrow y=1put values in ogn (2)\begin{array}{l}1=\frac{-0}{5}+\left(\frac{-1}{25}\right)+c \\c=1+\frac{1}{25}=\frac{26}{25} \\c=\frac{26}{25}\end{array}Hence\begin{array}{l}y=-\frac{x}{5}-\frac{1}{25}+\frac{26}{25} \\y=-\frac{x}{5}+\frac{25}{25} \\y=-\frac{x}{5}+1 \\y(x)=-\frac{x}{5}+1\end{array} ...