Consider the initial value problem below. The value of y at x = In 2, (rounded off to 3 decimal places) is \(\frac{{dy}}{{dx}} = 2x - y,\;y\left( 0 \right) = 1\)
Concept:
The standard form of a first-order linear differential equation is,
\(\frac{{dy}}{{dx}} + Py = Q\)
Where P and Q are the functions of x.
Integrating factor, \(IF = {e^{\smallint Pdx}}\)
Now, the solution for the above differential equation is,
\(y\left( {IF} \right) = \smallint IF.Qdx\)
Calculation:
\(\frac{{dy}}{{dx}} = 2x - y,\;y\left( 0 \right) = 1\)
\( \Rightarrow \frac{{dy}}{{dx}} + y = 2x\)
By comparing the above differential equation with the standard differential equation,
P = 1, Q = 2x
Integrating factor, \(IF = {e^{\smallint Pdx}} = {e^{\smallint 1dx}} = {e^x}\)
Now, the solution is
\(y\left( {{e^x}} \right) = \smallint {e^x}\left( {2x} \right)dx\)
\(y{e^x} = 2\left( {x{e^x} - {e^x}} \right) + C\)
\( \Rightarrow y = 2\left( {x - 1} \right) + C{e^{ - x}}\)
y(0) = 1
⇒ 1 = 2 (0 – 1) + C
⇒ C = 3
Now, the solution becomes
y = 2x – 2 + 3e-x
At x = ln 2,
y = 2 ln 2 – 2 + 3 (0.5) = 0.8862
Important Questions from First Order Equations
For the equation \(\frac{{dy}}{{dx}} + 7{x^2}y = 0\) , if y(0) = \(\frac{{3}}{{7}}\) , then the value of y(1) is
The differential equation \(\frac{{dy}}{{dx}} + 4y = 5\) is valid in the domain 0 ≤ x ≤ 1 with y (0) = 2.25 The solution of the differential equation is
The derivative of f(x) = cos(x) can be estimated using the approximation \(f'\left( x \right) = \frac{{f\left( {x + h} \right) - f\left( {x - h} \right)}}{{2h}}\) . The percentage error is calculated as \(\left( {\frac{{Exact\;value - Approximate\;value}}{{Exact\;value}}} \right) \times 100\). The percentage error in the derivative of f(x) at x = π/6 radian, choosing h = 0.1 radian, is
The general solution of the differential equation \(\frac{{dy}}{{dx}} = \cos \left( {x + y} \right)\), with c as a constant, is
Which one of the following is the general solution of the first order differential equation
\(\frac{{dy}}{{dx}} = {\left( {x + y - 1} \right)^2}\) , where x, y are real?
