Question

Find the general solution of the differential equation ​

Answer 1

Given Question :-

Find the general solution of the differential equation :

$\rm \: \dfrac{dy}{dx} = \dfrac{5x + 3}{2y + 7}$

$\large\underline{\sf{Solution-}}$

Given differential equation is

$\rm \: \dfrac{dy}{dx} = \dfrac{5x + 3}{2y + 7}$

On separating the variables, we get

$\rm \: (2y + 7)dy \: = \: (5x + 3)dx$

On integrating both sides, we get

$\rm \: \displaystyle\int\rm (2y + 7)dy \: = \: \displaystyle\int\rm (5x + 3)dx$

We know,

$\boxed{ \rm{ \:\displaystyle\int\rm {x}^{n} = \frac{ {x}^{n + 1} }{n + 1} + c \: }}$

and

$\boxed{ \rm{ \:\displaystyle\int\rm k \: dx \: = \: kx \: + \: c \: }}$

So, using these results, we get

$\rm \: 2 \times \dfrac{ {y}^{2} }{2} + 7y = 5 \times \dfrac{ {x}^{2} }{2} + 3x + c$

$\rm \: {y}^{2} + 7y = \dfrac{5{x}^{2} }{2} + 3x + c$

can be further rewritten as on multiply both sides by 2,

$\rm \: 2{y}^{2} + 14y = {5x}^{2}+6x + 2c$

$\rm \: 2{y}^{2} + 14y = {5x}^{2}+6x + d \: \: \: \{where \: d = 2c \}$

Hence, general solution of differential equation

$\rm \: \dfrac{dy}{dx} = \dfrac{5x + 3}{2y + 7} \: is \: \boxed{ \rm{ \:{2y}^{2} + 14y = {5x}^{2} + 6x + d \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$