Differential Equations Answers

y''-8y'+20y=100x²-26xe^x solve the given differential equation by undetermined cofficient

An inductor of 2 henries, resistor of 16 ohms and capacitor of 0.02 farads are connected in series with a battery of

e.m.f E = 100sin33t. At t=0, the charge on the capacitor and current in the circuit are zero. Find the charge and

current at time t.

J. A spring with a mass of 2 kg has natural length m. A force of 25.6 N

For a given set of constants α, β, and γ the functions

ƒ (x, y) =

7𝑥

𝛼𝑦

2𝑥𝛽

and ġ (x, y) =

3𝑥

𝛽−𝛾

𝑥𝛽−𝑦2

are homogeneous of degree 2 and 1 respectively. Determine the values for α, β, and γ.

An inductor of 2 henries, resistor of 16 ohms and capacitor of 0.02 farads are connected in series with a battery of

e.m.f E = 100sin33t. At t=0, the charge on the capacitor and current in the circuit are zero. Find the charge and

current at time t.

J. A spring with a mass of 2 kg has natural length m. A force of 25.6 N

Tank A initially contains 200, litres of brine containing 225 N of salt. Eight litres of fresh water per A and the mixture, assumed uniform, passes from A to b. initially .containing 200 litres of fresh water, at 8 litres per minute. The resulting mixture, also kept uniform, leaves B at the rate of 8 litres/min. Find the amount of minute enter salt in tank B after one hour.

Solve the linear partial differential equation (D⁴+D'⁴)=0

(i) State the Existence and Uniqueness theorem for the

differential equation of the first order.

(ii) A home buyer can spend no more than $700 per month on

mortgage payments. Suppose that the interest rate is

7% and that the term of the mortgage is 30 years.

Assume that the interest is compounded continuously

and that payments are also made continuously.

i. Determine the maximum amount that this buyer can

borrow.

ii. Determine the total interest paid during the term

of the mortgage

Verify that the function 𝑦 = 𝑐1𝑒

(−𝑘+2𝑖)𝑥 + 𝑐2𝑒

(−𝑘−2𝑖)𝑥 is a

solution to

𝑦

′′ + 2𝑘𝑦

′ + (𝑘

2 + 4)𝑦 = 0

Solve the following boundary value problems.

(i) 𝑦

′′ + 4𝑦 = 0; 𝑦(0) = 3, 𝑦(𝜋/2) = −3,

(ii) 𝑦

′′ − 25𝑦 = 0; 𝑦(−2) = 𝑦(2) = cosh 10.

(iii) 𝑦

′′ + 2𝑦

′ + 2𝑦 = 0; 𝑦(0) = 1, 𝑦(𝜋/2) = 0.

With the use of reduction of order for differential

equations, reduce the following to first order and

solve.

(i) 𝑦

′′ + 𝑒

𝑦𝑦

′3 = 0,

(ii) 𝑥𝑦

′′ + 2𝑦

′ + 𝑥𝑦 = 0, 𝑦1 =

sin 𝑥

𝑥

,

(iii) (1 − 𝑥

2

)𝑦

′′ − 2𝑥𝑦

′ + 2𝑦 = 0, 𝑦1 = 𝑥,

(iv) 4𝑥

2𝑦

′′ − 3𝑦 = 0, 𝑦(1) = 3, 𝑦

′

(1) = 2.5