Question

Solve the given differential equation by using appropriate Substitution;

$\frac{dy}{dx}=\frac{3x+2y}{3x+2y+2},y\left(-1\right)=-1$

Short answer

We will solve the given differential equation by substituting.

Step-by-step solution

Step 1:

Often the first step in solving the differential equation consists of transforming it into another differential equation by means of a substantial.

For example: Suppose we wish to transform the first order differential equation dy/dx=f(x,y) by the substitution y=g(x,u), where u is regarded as a function of the variable x.

Step 2:

Use the following substitution ;

$$\begin{gathered} u=3x+2y\to dy/dx=1/2\left(du/dx-3\right) \\ 1/2\left(du/dx-3\right)=u/u+2 \\ du/dx=\left(2u/u+2\right)+3=5u+6/u+2 \\ \left(u+2/5+6\right)du/dx=dx \end{gathered}$$

Integrate :

$$\begin{gathered} \int \left[1/5+4/5\left(5u+6\right)\right]du/dx=\int dx \\ u/5+4/251n\left|5u+6\right|=x+C_0 \\ 5u+41n\left|5u+6\right|=25x+C_1 \\ 5\left(3x+2y\right)+41n\left|5\left(3x+2y\right)+6\right|-25x=C_1 \\ 10\left(y-x\right)+41n\left|15x+10y+6\right|=C_1 \\ 5\left(y-x\right)+21n\left|15x+10y+6\right|=C \end{gathered}$$

Step 3:

Now we can use initial condition to find C. Did you perhaps forget to use find C.

$$\begin{gathered} C=5\left(-1+1\right)+21n\left|-15-10+6\right|=21n\left(19\right) \\ 5\left(y-x\right)+21n\left|15x+10y+6\right|=21n\left(19\right) \end{gathered}$$

Hence the final answer is$5\left(y-x\right)+21n\left|15x+10y+6\right|=21n\left(19\right)$