Solve the differential equation (2x-1)dx+(3y+7)dy=0

\left(2x-1\right)\cdot dx+\left(3y+7\right)\cdot dy=0

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Answer

$7y+\frac{3}{2}y^2-x+x^2=0+C_0$

Step by step solution

Problem

$\left(2x-1\right)\cdot dx+\left(3y+7\right)\cdot dy=0$
1

Grouping the terms of the differential equation

$\left(7+3y\right)dy=-\left(2x-1\right)dx$
2

Integrate both sides, the left side with respect to $y$, and the right side with respect to $x$

$\int\left(7+3y\right)dy=\int-\left(2x-1\right)dx$
3

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int7dy+\int3ydy=\int-\left(2x-1\right)dx$
4

The integral of a constant is equal to the constant times the integral's variable

$7y+\int3ydy=\int-\left(2x-1\right)dx$
5

Taking the constant out of the integral

$7y+\int3ydy=-\int\left(2x-1\right)dx$
6

The integral of a sum of two or more functions is equal to the sum of their integrals

$7y+\int3ydy=-\left(\int-1dx+\int2xdx\right)$
7

The integral of a constant is equal to the constant times the integral's variable

$7y+\int3ydy=-\left(\int2xdx-x\right)$
8

Taking the constant out of the integral

$7y+\int3ydy=-\left(2\int xdx-x\right)$
9

Taking the constant out of the integral

$7y+3\int ydy=-\left(2\int xdx-x\right)$
10

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$7y+3\int ydy=-\left(2\cdot \frac{1}{2}x^2-x\right)$
11

Multiply $\frac{1}{2}$ times $2$

$7y+3\int ydy=-\left(1x^2-x\right)$
12

Any expression multiplied by $1$ is equal to itself

$7y+3\int ydy=-\left(x^2-x\right)$
13

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$7y+3\cdot \frac{1}{2}y^2=-\left(x^2-x\right)$
14

Multiply $\frac{1}{2}$ times $3$

$7y+\frac{3}{2}y^2=-\left(x^2-x\right)$
15

Multiply $\left(x^2+-x\right)$ by $-1$

$7y+\frac{3}{2}y^2=x-x^2$
16

Rewrite the equation

$-\left(x-x^2\right)+7y+\frac{3}{2}y^2=0$
17

Multiply $\left(-x^2+x\right)$ by $-1$

$7y+\frac{3}{2}y^2-x+x^2=0$
18

Add the constant of integration

$7y+\frac{3}{2}y^2-x+x^2=0+C_0$

Answer

$7y+\frac{3}{2}y^2-x+x^2=0+C_0$

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Problem Analysis

Main topic:

First order differential equations

Time to solve it:

0.5 seconds

Views:

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