# Find the general solution of the given differential

Find the general solution of the given differential equations
$4y{}^{″}-4{y}^{\prime }-3=0$
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Cassius Villarreal
Given:
$4y{}^{″}-4{y}^{\prime }-3=0$
Concept:
The non ­- homogenous ODE, second-order linear equation of type $ay{}^{″}+b{y}^{\prime }$=constant
$y={y}_{h}+{y}_{p}$ is the general solution of $ay{}^{″}+b{y}^{\prime }$=constant
The solution:
${y}_{h}$ is the solution to the homogeneous ODE $ay{}^{″}+b{y}^{\prime }=0$
${y}_{p}$ the particular solution, is any function that satisfies the non-homogeneous equation
Solution:
$4y{}^{″}-4{y}^{\prime }=0$
The characteristic equation:
$4{m}^{2}-4m=0$
$4m\left(m-1\right)=0$
$m=0$ and $m=1$
The solution when roots are $a=0$ and $b=0$ is ${c}_{1}{e}^{ax}+{c}_{2}{e}^{bx}$
${y}_{h}={c}_{1}{e}^{1x}+{c}_{2}{e}^{\left(0\right)x}$
${y}_{h}={c}_{1}{e}^{x}+{c}_{2}{e}^{0}$
${y}_{h}={c}_{1}{e}^{x}+{c}_{2}$
Now,
${y}_{p}=-\frac{3}{4}t$
so,
$y={y}_{h}+{y}_{p}$
$y=\left({c}_{1}{e}^{x}\right)+{c}_{2}\right)+\left(-\frac{3t}{4}\right)$
$y={c}_{1}{e}^{x}+{c}_{2}-\frac{3t}{4}$
$y={c}_{1}{e}^{x}+{c}_{2}-\frac{3t}{4}$
Jeffrey Jordon