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rosijanka [135]
3 years ago
10

You decide to borrow money to pay for college at 22% interest per year, in 2023.

Mathematics
1 answer:
Anastaziya [24]3 years ago
3 0

Answer: On  loan of $5000 at that rate, you would be owing $9400

Step-by-step explanation: The loan is being borrowed at an interest rate that is calculated yearly. This is a case of a simple interest. The same amount of interest is calculated and that will be payable per year. The amount borrowed is the principal, and in this instance is $5000. The amount of interest payable is calculated as 22 percent of the principal and is paid yearly. So if a loan (principal amount) of $5000 is borrowed for a period of 4 years (that is, 2023 to 2027), then the interest is calculated for four years.

Hence, the formula for a simple interest computation is given as follows;

Interest = P x R x T

Where P = 5000, Rate = 0.22 (22 percent), and T = 4 (time in years)

Interest = 5000 x 0.22 x 4

Interest = 20000 x 0.22

Interest = 4400

Therefore, from the calculations shown, a loan of $5000 at the rate of 22% borrowed from 2023 to 2027 would have an interest payable of $4,400.

Upon repayment of the loan, you would be owing a total of $9,400 (that is, amount borrowed plus interest owing)

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Solve the nonhomogeneous differential equation y′′+25y=cos(5x)+sin(5x). Find the most general solution to the associated homogen
Marrrta [24]

Answer:

y(x)=c_1cos(5x)+c_2sin(5x)+0.1xsin(5x)-0.1xcos(5x)

Step-by-step explanation:

The general solution will be the sum of the complementary solution and the particular solution:

y(x)=y_c(x)+y_p(x)

In order to find the complementary solution you need to solve:

y''+25y=0

Using the characteristic equation, we may have three cases:

Real roots:

y(x)=c_1e^{r_1x} +c_2e^{r_2x}

Repeated roots:

y(x)=c_1e^{rx} +c_2xe^{rx}

Complex roots:

y(x)=c_1e^{\lambda x}cos(\mu x) +c_2e^{\lambda x}sin(\mu x)\\\\Where:\\\\r_1_,_2=\lambda \pm \mu i

Hence:

r^{2} +25=0

Solving for r :

r=\pm5i

Since we got complex roots, the complementary solution will be given by:

y_c(x)=c_1cos(5x)+c_2sin(5x)

Now using undetermined coefficients, the particular solution is of the form:

y_p=x(a_1cos(5x)+a_2sin(5x) )

Note: y_p was multiplied by x to account for cos(5x) and sin(5x) in the complementary solution.

Find the second derivative of y_p in order to find the constants a_1 and a_2 :  

y_p''(x)=10a_2cos(5x)-25a_1xcos(5x)-10a_1sin(5x)-25a_2xsin(5x)

Substitute the particular solution into the differential equation:

10a_2cos(5x)-25a_1xcos(5x)-10a_1sin(5x)-25a_2xsin(5x)+25(a_1xcos(5x)+a_2xsin(5x))=cos(5x)+sin(5x)

Simplifying:

10a_2cos(5x)-10a_1sin(5x)=cos(5x)+sin(5x)

Equate the coefficients of cos(5x) and sin(5x) on both sides of the equation:

10a_2=1\\\\-10a_1=1

So:

a_2=\frac{1}{10} =0.1\\\\a_1=-\frac{1}{10} =-0.1

Substitute the value of the constants into the particular equation:

y_p(x)=-0.1xa_1cos(5x)+0.1xsin(5x)

Therefore, the general solution is:

y(x)=y_c(x)+y_p(x)

y(x)=c_1cos(5x)+c_2sin(5x)+0.1xsin(5x)-0.1xcos(5x)

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Step-by-step explanation:

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