1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Gre4nikov [31]
3 years ago
15

can someone show me how to find the general solution of the differential equations? really need to know how to do it for the upc

oming exam

Mathematics
1 answer:
mariarad [96]3 years ago
6 0
The first equation is linear:

x\dfrac{\mathrm dy}{\mathrm dx}-y=x^2\sin x

Divide through by x^2 to get

\dfrac1x\dfrac{\mathrm dy}{\mathrm dx}-\dfrac1{x^2}y=\sin x

and notice that the left hand side can be consolidated as a derivative of a product. After doing so, you can integrate both sides and solve for y.

\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1xy\right]=\sin x
\implies\dfrac1xy=\displaystyle\int\sin x\,\mathrm dx=-\cos x+C
\implies y=-x\cos x+Cx

- - -

The second equation is also linear:

x^2y'+x(x+2)y=e^x

Multiply both sides by e^x to get

x^2e^xy'+x(x+2)e^xy=e^{2x}

and recall that (x^2e^x)'=2xe^x+x^2e^x=x(x+2)e^x, so we can write

(x^2e^xy)'=e^{2x}
\implies x^2e^xy=\displaystyle\int e^{2x}\,\mathrm dx=\frac12e^{2x}+C
\implies y=\dfrac{e^x}{2x^2}+\dfrac C{x^2e^x}

- - -

Yet another linear ODE:

\cos x\dfrac{\mathrm dy}{\mathrm dx}+\sin x\,y=1

Divide through by \cos^2x, giving

\dfrac1{\cos x}\dfrac{\mathrm dy}{\mathrm dx}+\dfrac{\sin x}{\cos^2x}y=\dfrac1{\cos^2x}
\sec x\dfrac{\mathrm dy}{\mathrm dx}+\sec x\tan x\,y=\sec^2x
\dfrac{\mathrm d}{\mathrm dx}[\sec x\,y]=\sec^2x
\implies\sec x\,y=\displaystyle\int\sec^2x\,\mathrm dx=\tan x+C
\implies y=\cos x\tan x+C\cos x
y=\sin x+C\cos x

- - -

In case the steps where we multiply or divide through by a certain factor weren't clear enough, those steps follow from the procedure for finding an integrating factor. We start with the linear equation

a(x)y'(x)+b(x)y(x)=c(x)

then rewrite it as

y'(x)=\dfrac{b(x)}{a(x)}y(x)=\dfrac{c(x)}{a(x)}\iff y'(x)+P(x)y(x)=Q(x)

The integrating factor is a function \mu(x) such that

\mu(x)y'(x)+\mu(x)P(x)y(x)=(\mu(x)y(x))'

which requires that

\mu(x)P(x)=\mu'(x)

This is a separable ODE, so solving for \mu we have

\mu(x)P(x)=\dfrac{\mathrm d\mu(x)}{\mathrm dx}\iff\dfrac{\mathrm d\mu(x)}{\mu(x)}=P(x)\,\mathrm dx
\implies\ln|\mu(x)|=\displaystyle\int P(x)\,\mathrm dx
\implies\mu(x)=\exp\left(\displaystyle\int P(x)\,\mathrm dx\right)

and so on.
You might be interested in
FjItskgzgxllxzylgzlgzlgzllzlzlz
Arlecino [84]

Answer:

fncbcghdhdgdgdnf thats your answer

Step-by-step explanation:

4 0
2 years ago
Write the fact family for 5,8,and 40
Juli2301 [7.4K]
A fact family is basically just different equations with the same numbers getting the same/different numbers as answers you might say... 
this will make more sense: 
5x8=40
8x5=40
40/8=5
40/5=8
4 0
2 years ago
Write an expression for the product of 11 and w
Nadya [2.5K]

Answer:

11 * w

Step-by-step explanation:

"product" in math means "multiply". "The product of 11 and w" is "the multiplication of 11 and w".

Write the expression:

The Multiplication: *

11 and w: 11 * w

11 * w is your answer.

~

3 0
2 years ago
You and 3 friends decided to share a pizza. If everyone got 2 pieces each, how many total pieces of pizza were there to begin wi
Musya8 [376]

Answer:

8

Step-by-step explanation:

dhdhdhdhdhdhdhddijd

6 0
2 years ago
Read 2 more answers
Can you Graph the image A(4,-1) after a translation 7 units up?
rusak2 [61]
(4,6) is the correct answer
4 0
2 years ago
Other questions:
  • when a decimal is written in word form, what indicates that the equivalent form is a mixed number and not a fraccion? Explain.
    11·1 answer
  • Please help and thank you
    15·1 answer
  • PLEASE HELP ME
    5·2 answers
  • Tell wheather the system has one solution, infinitely many solutions, or no solution.
    9·1 answer
  • What is the wquation of a circle with center (-2.3) and radius 4
    5·1 answer
  • The weights of a small box of healthy oats are normally distributed with a mean of 8.9 oz and a standard deviation of 0.1 oz. Fi
    7·2 answers
  • 5. Stacy bought 4 trays of eggs at sh.150 per tray.
    12·1 answer
  • An ice-cream store sold 70 sundaes. The ratio of
    6·1 answer
  • Big money butts how to do.​
    10·1 answer
  • What is the area of a regular hexagon with a side length of 4 m? enter your answer in the box. round only your final answer to t
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!