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3x + 7 Find the particular solution for the equation given y = 4 when x = 0. dx y-5The solution to the differential equation is y2 (Type an expression using xand y ...

Question

3x + 7 Find the particular solution for the equation given y = 4 when x = 0. dx y-5The solution to the differential equation is y2 (Type an expression using xand y as the variables. Use integers or fractions for any numbers in the expression )

3x + 7 Find the particular solution for the equation given y = 4 when x = 0. dx y-5 The solution to the differential equation is y2 (Type an expression using xand y as the variables. Use integers or fractions for any numbers in the expression )



Answers

Find the general or particular solution, as indicated, for each differential equation. $$ \frac{d y}{d x}=\frac{7}{x} $$

Remember that these questions can be put into a linear form over why, plus p of acts. This'd why Prime plus p of X y is equal to queue of acts, so some algebra algebraic manipulation may be required in order to particular put it into this particular form. So here we have outlined the steps and found why for the general's new ships were defined P of x two equal to divided by X and cube off X to be three x minus five. So therefore you of X is equal to eat the power of the integral off P of X d x or you of axes equal t to the power of the integral of two. Divided by X t x On the interval of two divided by ex the ex is e to the power off the Ellen of X squared, which is just equal to X square. Next we have why is equal to one over you of X things eat the power of the integral off you of X times que of acts de axe. So this is equal to one over X squared E to the power of the integral of X square Times three x minus five This is X square times three X minus five The integral And if we solve this out, we get that. Why, yeah, Why is equal to three x squared, Divided by four minus five x minus five x Divided by three plus, of course, see divided by X square. So if we solve for awhile prime, we get why crime is equal to three acts divided by two minus five over three minus two seed divided by ex cute. So the only way for sure to know that your steps are correct is to prove that your answer fits into theoretical differential equation. So we have D Y. The X Plus two over X times Why is equal to three X minus five? So we solved for D Y the X right here. That's why prime and resell for Why over here So plugging this that three x over to minus five over three minus two C divided by ex huge plus two over x times three X squared over four minus five x divided by three plus C over X squared is equal to three x minus five. And if you solve this out after simple algebra, you're gonna get that. Yes, it is, in fact, the correct solution

All right, So we're following 25. We have to find the general solution to this differential equation. And when you have a simple differential equation like when all the coefficients are constants, you can assume that the solution this in the form of each of the power of K of a constant times, the independent variable, which I'm just gonna make it out to be us X. And since that this is gonna be the form of our solution, we just a different differentiated. So it's gonna be a wine primacy, cause I k time U to the power of K X or a t of whatever you choose for your independence Very well. And why double crime's gonna be case Grace, he goes, No. Well, hurry Dan's case, Great time Q to the power of K acts. And then we just a substitute these into the differential equation. We want to solve so something. Three times Case crew time to eat of hurricane acts minus two times K times used to the parquet access minus seven times using fire K X because zero, we factor out e to power chaotic. So it's gonna be each of the power K acts times in three K squared minus two caring when this seven issue with zero. Now I have to find the values of K such that this functional equals zero. Well, we know the left. A partisan is a on exponential function, which means is never gonna call zero, which means we're going to have to rely on this. The rights part c equals zero, and conveniently, it's justic quadratic equation. So it's gonna be three k squared, minus two came on the seven years egos of zero. We used the quadratic formula, so it's gonna be not gonna be or positive to Post online is negative square, which is for minus four times three times, I guess. Seven all over. It's your time Street. So it's give me two plus or minus the square root of four for us 84 since the four times three is 12 and then times seven, which is 84 the negatives cancel on and on the volume we have sex. So it's going to put reminds the square root of 88 over sex. We're going to rewrite the square root of 88 ass serious groups here. So it's going to force Reminds to spirit of 22 Oh, her sex and we simplifying. So it's gonna give us one false reminds the square root of 22/3. And just like that, we found the solutions for Kay and the solutions for this differential equations. So it's gonna be why is he going Teoh a constant times U to the power of one. Minus the script are the 22 or three acts and we added the solutions together. So it's gonna be plus another constant times each of the power one plus the square root of 22/3 X and, yeah, that's basically X.

We need to solve the following equation both for X. And then we need to solve it again for why? So our first one we're gonna solve for X. So we're gonna move the four way over to the other side. So we have three X. Four wives cancel out equals we're gonna put it in order variable part first. So four Y plus seven four Y. And seven or not friends. So we can't combine them. They're gonna divide each term by three and you get X. Is equal to four thirds times Y plus seven thirds. Our second one we're going to solve for Y. So we're gonna move our three X over. So our three X's cancel out and we're left with negative four. Y. Don't forget to bring the negative sign that was with the four equals negative three X plus seven. Again three X. And seven are not common terms. We can't combine them. You're going to divide each term by a negative four because you want a positive Y. So we get Y. Is equal to positive 3/4 times X minus a minus 7/4.

Hi. This is a video on how to solve the differential equation. Why Prime Plus Three. Why is equal to E to the three X now? This is example of the first order Lydia Differential equation, and so it would be solved using the method of integrating factors as well Do now. Now a General First Order linear differential equation can be written in the form of why prime plus p of X. Why it was your ex with some functions. P and can cute here. P of X is equal to three. Q x is equal to the E to the three X. Now in general, the method of integrating factor starts by finding an integrating factor You, which is a function of X given by E to the anti derivative off P now hit P is equal to three, and the anti derivative of three is three x, So the integrating factor here is each of the three X. Now, the results on interest integrating factors says that a general solution to the differential equation given here is if the form one over you have x times the integral Q of X, you of X t X now here you have X is equal to eat up three x so we can write one over. You have access key to the negative three X and now we're integrating Q rex, which it is e to the three x times you of x, which is also you to the three X. So just moving to another screen here you have. Why is equal to eat it? Negative three Ex integral off due to the three x you to the three x, which is equal to e to the native three x into girl E to the six x Now the anti derivative of into the six x is e to the six x of the six, plus some arbitrary, constant C. So now if we simplify this equation, we're left with e to the three x over six plus c you to the native three x That is the answer


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