First Order ODEs
We will cover following topics
Formulation Of Differential Equations
A differential equation (DE) is a relation that contains a finite set of functions and their derivatives with respect to one or more independent variables.
An ordinary differential equation (ODE) is a relation that contains derivatives of functions of only one variable.
The highest derivative that appear in a ODE is the order of that ODE.
If the ODE can be written as a0(x)y(n)(x)+a1(x)y(n−1)(x)+⋯+an−1(x)y(1)(x)+an(x)y(x)=F(x), then the given ODE is linear.
If such representation is not possible, then we say that the given ODE is nonlinear.
Equations Of First Order And First Degree
Method of Variation of Parameters: Consider the equation dydx+P(x)y=Q(x).
The homogeneous solution is calculated by putting Q(x)=0 in the given differential equation.
Solving for y, we get y=AeP(x).
Now, according to the method of variation of parameters, we write the solution as:
y=v(x)eP(x) where A is replaced by v(x).
The complete solution is written as:
y=(v(x)+A)e−∫P(x)where v(x)=eP(x), v′(x)=Q(x)e−∫P(x)
Separable Equations: These are equations of the forms y′=f(x)g(y) and can be solved as ∫dyg(y)=∫f(x)dx+C, provided g(y)≠0.
If an ODE is of the form y′=F(ax+by+c), it can be reduced to the separable form by substituting ax+by+c=v.
Similarly, if the ODE is of the form y′=f(y/x), it can be reduced to the separable form by substituting v=y/x.
Exact Equation: A first order ODE of the form M(x,y)dx+N(x,y)dy=0 is exact if there exists a function u(x,y) such that
M=∂u∂x and N=∂u∂y, so that the given ODE can be written as du=0 and the solution becomes u=c.
Theorem: Let M and N be defined and continuously differentiable on a rectangle R={(x,y):|x−x0|<a,|y−y0|<b}.
Then Mdx+Ndy=0 is exact if and only if:
∂M/∂y=∂N/∂x ∀(x,y)∈RIntegrating Factor
Integrating Factor*: An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable
Exact Differential Equation: The differential equation Mdx+Ndy=0 is said to be exact if My=Nx.
If the equation is not exact, it can be made exact by multiplying the entire equation by μ(x,y) such that the resultant equation (μM)dx+(μN)dy=0. Here, μ is called the integrating factor.
Two special cases arise when the given equation is not exact, i.e. My≠Nx:
Case 1: If (My−NxN) is a function of x only, say ξ(x). Then, the integrating factor is given by:
μ(x)=e∫ξ(x)dxCase 2: If (My−Nx−M) is a function of y only, say ψ(y). Then, the integrating factor is given by:
μ(y)=e∫ψ(y)dyOrthogonal Trajectory
The orthogonal trajectory for a given curve y=f(x,c) can be found by forming the differential equation (eliminating c) and replacing dy/dx by −1dy/dx
Equations Of First Order But Not Of First Degree
Such differential equations contain only the first order differential coefficient dydx but will occur in a degree higher than one.
These equations can be solved by factorization and solving for dydx.
Clairaut’s Equation and Singular Solution
An equation of the form y=px+f(p), where p=dydx is called Clairaut’s Equation.
Differentiating this equation, we get p′[x+f′(p)]=0, which implies that either p′=0, or x+f′(p)=0.
Therefore, the first equation p′=0 gives the general solution of the Clairaut’s Equation as y=cx+f(c) and the second equation x+f′(p)=0, combined with the Clairaut’s Equation y=px+f(p), gives the singular solution (by eliminating p) which is free from any arbitrary constant.
PYQs
Formulation Of Differential Equations
1) Find the differential equation representing the entire circle in the xy−plane.
[2017, 10M]
2) Consider the differential equation y′=αy,x>0 where α is a constant. Show that:
i) If ϕ(x) is any solution and ψ(x)=ϕ(x)e−αx, then ψ(x) is a constant.
ii) If α<0, then every solution tends to zero as x→∞.
[2010, 12M]
3) Find the differential equation of the family of circles in the xy−plane passing through (−1,1) and (1,1).
[2009, 20M]
4) A continuous function y(t) satisfies the differential equation:
dydx={1+e1−t,0≤t<12+2t−3t2,1≤t<5If y(0)=−e, find y(2).
[2001, 12M]
Equations Of First Order And First Degree
1) Determine the complete solution of the differential equation
d2ydx2−4dydx+4y=3x2e2xsinx[2019, 10M]
2) Solve the differential equation
(2ysinx+3y4sinxcosx)dx−(4y3cos2x+cosx)dy=0[2019, 10M]
3) Solve the following simultaneous liner differential equations:
(D+1)y=z+ex and (D+1)z=y+ex
where y and z are functions of independent variable x and D≡ddx.
[2017, 8M]
4) If the growth rate of the population of bacteria at time t is proportional to the amount present at the time t and population doubles in one week, then how much bacteria can be expected after 4 weeks?
[2017, 8M]
5) Solve:
dydx=11+x2(etan−1x−y)[2016, 10M]
6) Solve the differential equation:
(2xy4ey+2xy3+y)dx+(x2y4ey−x2y2−3x)dy=0.
[2015, 10M]
7) Solve by the method of variation of parameters: dydx−5y=sinx.
[2014, 10M]
8) If y is a function of x such that the differential coefficient dydx is equal to cos(x+y)+sin(x+y), find out a relation between x and $y$$ which is free from any derivative/ differential.
[2013, 10M]
9) Solve the differential equation (5x3+12x2+6y2)dx+6xydy=0
[2013, 15M]
10) Solve dydx=2xye(x/y)2y2(1+e(x/y)2)+2x2e(x/y)2.
[2012, 12M]
11) Obtain the solution of the ordinary differential equation dydx=(4x+y+1)2, if y(0)=1
[2011, 10M]
12) Show that the set of solutions of the homogeneous linear differential equation
y′+p(x)y=0on an interval I=[a,b] forms a vector subspace W of the real vector space of continuous functions on I. What is the dimension of W?
[2010, 20M]
13) Solve: dydx=y2(x−y)3xy2−x2y−4y3,y(0)=1
[2009, 20M]
14) Solve the differential equation ydx+(x+x3y2)dy=0
[2008, 12M]
15) Find the solution of the equation dyy+xy2dx=−4xdx
[2007, 12M]
16) Solve the differential equation: (xy2+e−1x3)dx−x2ydy=0.
[2006, 12M]
17) Solve: (1+y2)+(x−e−tan−1y)dydx=0.
[2006, 15M]
18) Solve: xydydx=√(x2−y2−x2y2−1).
[2005, 12M]
19) Solve: y(xy+2x2y2)dx+x(xy−x2y2)dy=0.
[2004, 12M]
20) Solve: xdydx+ylogy=xyex.
[2003, 12M]
21) Solve: xdydx+3y=x3y2.
[2002, 12M]
22) Solve: dydx=x+y+4x−y−6.
[2002, 15M]
23) Solve: dydx+yxlogey=y(logey)2x2.
[2001, 15M]
Integrating Factor
1) Find f(y) such that (2xey+3y2)dy+(3x2+f(y))dx=0 is exact and hence solve.
[2018, 13M]
2) Find α and β such that xαyβ is an integrating factor of (4y2+3xy)dx−(3xy+2x2)dy=0 and solve the equation.
[2018, 12M]
3) Solve {y(1−xtanx)+x2cosx}dx−xdy=0.
[2016, 10M]
4) Solve the differential equation: xcosxdydx+y(xsinx+cosx)=1.
[2015, 10M]
5) Find the constant a so that (x+y)a is the integrating factor of (4x2+2xy+6y)dx+(2x2+9y+3x)dy=0 and hence solve the differential equation.
[2015, 12M]
6) Justify that a differential equation of the form: [y+xf(x2+y2)]dx+[yf(x2+y2)−x]dy=0 where f(x2+y2) is an arbitrary function of (x2+y2), is not an exact differential equation and 1x2+y2 is an integrating factor for it. Hence solve this differential equation for f(x2+y2)=(x2+y2)2.
[2014, 10M]
7) Find the sufficient condition for the differential equation M(x,y)dx+N(x,y)dy=0 to have an integrating factor as a function of (x+y). What will be the integrating factor in that case? Hence find the integrating factor for the differential equation of (x2+xy)dx+(y2+xy)dy=0 and solve it.
[2014, 15M]
8) Show that the differential equation (2xylogy)dx+(x2+y2√y2+1)dy=0 is not exact. Find an integrating factor and hence, the solution of the equation.
[2012, 20M]
9) Show that the differential equation (3y2−x)+2y(y2−3)y′=0 admits an integrating factor which is a function of (x+y2). Hence solve the equation.
[2010, 12M]
10) Verify that 12(Mx+Ny)d[loge(xy)]+12(Mx−Ny)d[loge(x/y)]=Mdx+Ndy. Hence show that:
i) If the differential equation Mdx+Ndy=0 is homogeneous, then (Mx+Ny) is an integrating factor unless Mx+Ny≡0.
ii) If the differential equation Mdx+Ndy=0 is not exact but is of the form f1(xy)ydx+f2(xy)xdy=0 then (Mx−Ny)−1 is an integrating factor unless Mx+Ny≡0.
[2010, 20M]
11) Solve the ordinary differential equation cos3xdydx−3ysin3x=12sin6x+sin23x, 0<x<π2.
[2007, 15M]
12) Find the solution of the following differential equation dydx+ycosx=12sin2x.
[2004, 12M]
13) Find the value of constant λ such that the following differential equation becomes exact.
(2xey+3y2)dydx+(3x2+λey)=0. Further, for this value of λ, solve the equation.
[2002, 15M]
Orthogonal Trajectory
1) Show that the family of parabolas y2=4cx+4c2 is self orthogonal.
[2016, 10M]
2) Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency.
[2014, 10M]
3) Obtain the equation of the orthogonal trajectory of the family of curves represented by rn=asinnθ,(r,θ) being the plane polar coordinates.
[2013, 10M]
4) Find the orthogonal trajectories of the family of curves x2+y2=ax.
[2012, 12M]
5) Determine the orthogonal trajectory of a family of curves represented by the polar equation r=a(1−cosθ),(r,θ) being the plane polar coordinates of any point.
[2011, 10M]
6) Find the family of curves whose tangents form an angle π4 with the hyperbolas xy=c,c>0.
[2006, 12M]
7) Find the orthogonal trajectory of the family of co-axial circles x2+y2+2gx+c=0, where g is the parameter.
[2005, 12M]
8) Show that the orthogonal trajectory of a system of confocals ellipses is self orthogonal.
[2003, 12M]
Equations Of First Order But Not Of First Degree
1) Solve the differential equation x=py−p2 where p=dydx.
[2015, 13M]
2) Solve the equation y−2xp+yp2=0, where p=dydx.
[2008, 15M]
Clairaut’s Equation and Singular Solution
1) Solve:
(dydx)2y+2dydx−y=0[2018, 13M]
2) Consider the differential equation xyp2−(x2+y2−1)p+xy=0 where p=dydx substituting u=x2 and v=y2 reduce the equation to Clairaut’s form in terms of u, v and p′=dvdu hence or otherwise solve the equation.
[2017, 10M]
3) Obtain Clairaut’s form of the differential equation (xdydx−y)(ydydx+x)=a2dydx Also find its general solution.
[2011, 15M]
4) Determine the general and singular solutions of the equation
y=xdydx+adydx[1+(dydx)2]−12,a being a constant.
[2007, 15M]
5) Solve the equation
x2p2+yp(2x+y)+y2=0using the substitution y=u and xy=v and find its singular solution, where p=dydx
[2006, 15M]
6) Solve the differential equation: (x2+y2)(1+p)2−2(x+y)(1+p)(x+yp)+(x+yp)2=0 where p=dydx, by reducing it to Clairaut’s form by using suitable substitution.
[2005, 15M]
7) Reduce the equation (px−y)(py+x)=2p, where p=dydx to Clairaut’s equation and hence solve it.
[2004, 15M]
8) Solve the differential equation (px2+y2)(px+y)=(P+1)2, where p=aydx, by reducing it to Clairaut’s form using suitable substitutions.
[2003, 15M]