👉 Try now NerdPal! Our new math app on iOS and Android

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

Step-by-step Solution

Go!
Math mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Final Answer

$\frac{3}{2}y^2+7y=-x^2+x+C_0$
Got another answer? Verify it here!

Step-by-step Solution

Specify the solving method

1

The differential equation $\left(2x-1\right)dx+\left(3y+7\right)dy=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$

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

Find the derivative of $M(x,y)$ with respect to $y$

$\frac{d}{dy}\left(2x-1\right)$

The derivative of the constant function ($2x-1$) is equal to zero

0

Find the derivative of $N(x,y)$ with respect to $x$

$\frac{d}{dx}\left(3y+7\right)$

The derivative of the constant function ($3y+7$) is equal to zero

0
2

Using the test for exactness, we check that the differential equation is exact

$0=0$

Expand the integral $\int\left(2x-1\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int2xdx+\int-1dx$

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

$\int2xdx-x$

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

$2\int xdx-x$

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

$1x^2-x$

Any expression multiplied by $1$ is equal to itself

$x^2-x$

Since $y$ is treated as a constant, we add a function of $y$ as constant of integration

$x^2-x+g(y)$
3

Integrate $M(x,y)$ with respect to $x$ to get

$x^2-x+g(y)$

The derivative of the constant function ($x^2-x$) is equal to zero

0

The derivative of $g(y)$ is $g'(y)$

$0+g'(y)$
4

Now take the partial derivative of $x^2-x$ with respect to $y$ to get

$0+g'(y)$

Simplify and isolate $g'(y)$

$3y+7=0+g$

$x+0=x$, where $x$ is any expression

$3y+7=g$

Rearrange the equation

$g=3y+7$
5

Set $3y+7$ and $0+g'(y)$ equal to each other and isolate $g'(y)$

$g'(y)=3y+7$

Integrate both sides with respect to $y$

$g=\int\left(3y+7\right)dy$

Expand the integral $\int\left(3y+7\right)dy$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$g=\int3ydy+\int7dy$

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

$g=\int3ydy+7y$

The integral of a function times a constant ($3$) is equal to the constant times the integral of the function

$g=3\int ydy+7y$

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

$g=\frac{3}{2}y^2+7y$
6

Find $g(y)$ integrating both sides

$g(y)=\frac{3}{2}y^2+7y$
7

We have found our $f(x,y)$ and it equals

$f(x,y)=x^2-x+\frac{3}{2}y^2+7y$
8

Then, the solution to the differential equation is

$x^2-x+\frac{3}{2}y^2+7y=C_0$
9

Group the terms of the equation

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

Final Answer

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

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Linear Differential EquationExact Differential EquationSeparable Differential EquationHomogeneous Differential Equation

Give us your feedback!

Function Plot

SnapXam A2
Answer Assistant

beta
Got a different answer? Verify it!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

Your Math & Physics Tutor. Powered by AI

Available 24/7, 365.

Complete step-by-step math solutions. No ads.

Includes multiple solving methods.

Support for more than 100 math topics.

Premium features on our iOS and Android app.

Join 500k+ students in problem solving.

Subscription plan. Cancel anytime.
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account
3-Month Special Plan
One-time payment of $2.97 USD.
Without automatic renewal.
Create an Account