## Vineet Loomba is teaching live on Unacademy Plus

DIFFERENTIAL EqUATIONS (DETAILED COURSE) JEE MAIN AND ADVANCED IIT-IEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITiAN | IIT-JEE MENTOR FoR SURE SHoT SUcCESS IN JEE MAIN AND ADVANCED (IIT-JEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) ABOUT ME B.Tech. From IIT Roorkee -$ IIT-JEE Mentor Since 2010. -^ Doubts/Feedback in Comment Section * Comment other topics you want to revise. * Follow me @ https://unacademy.com/user/vineetloomba to get unacademy updates or search me on Google * Share among your peers as SHARING is CARING!!

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Other Detailed Courses Made so far on Unacademv: V Strategy for JEE Main and Advanced V Sets, Relations and Functions V Trigonometry V Applications of Derivatives Limits, Continuity. Differentiability V Indefinite Integration V Definite Integration Complex Number V Logarithmic Functions V Sequences Series V Most Important Questions in IIT-JEE Permutations Combinations Binomial Theorem V Straight Lines V Applications of Integrals V Circles (Detailed Course) V Probability MathematicS V Parabola (Detailed Course) v Inverse Trigonometry V Mathematical Reasoning V Ellipse (Detailed Course) V Hyperbola V Differential Equations Upcoming Courses next month: Statistics Matrices Determinants MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) An ORTHOGONAL TRAJECTORY of a given system of curves is defined as a curve which cuts every member of a given family of curve at right angle y =2x x +y-1 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (11T-JEE) Method to find orthogonal trajectory: (i) Letf (x, y, c) 0 be the equation of the given family of curves, (i) Differentate the given equation w.r.t. x and then eliminate c. iReplace dx y- dy in the equation obtained in step (i). v) Solve the differential equation obtained in step (ii). Above solution is the required orthogonal trajectory. where'c' is an arbitrary constant. dx MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Example: Find the orthogonal trajectory of family of straight lines passing through the origin Solution. Family of straight lines passing through the origin is y = mx (1 where 'm' is an arbitrary constant dym Differentiating wrt x, we get dy Eliminate 'rm' from (i) & (ii) y =-x dx by dy dx we get y dy dy Replacing dx x dx y dy 0 2 Integrating which is the required orthogonal trajectory ER. VINEET LOOMBA (IIT RooRKee) MATHEMATICS FOR IIT-JEE

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Example: Find the orthogonal trajectories of the family of curves x + y-C- Solution: The family F(x, y, c)-x + -C2, consists of circles with centers at the origin and radii c dy x dy y dy' dx x Replace bv dX This equation is in variable separable form, its solution is y = kx which represents the orthogonal trajectories. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Example : Find the orthogonal trajectory of y2 Solution. y- 4ax 4ax (a being the parameter) 2 dy Eliminating 'a' from (i) & (ii) y2-2y x dx Replacing dx dy we get y-2 2 x dx y dy 0 2 Integrating, x2c 2x2 y2c which is the required orthogonal trajectories MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Example: Find the orthogonal trajectories of the family of curves y cx2 Solution: The family, which is given by F(x, y, c)-y-cx consists of parabola symmetric about the y-axis with vertices at the origin. Differentiating the given equation with respect to X, dy dx we obtain2CX Here f(x, y)- dx 2y or x dx + 2y dy :0 The solution of this separable equation is yk 1,2 1v2 2 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (11T-JEE) SELF PRACTICE PROBLEMS 1.Find the orthogonal trajectory of family of circles touching x - axis at the origin. Ans xy cx where c is an arbitrary constant. 2. Find the orthogonal trajectories of the family of curves y2 4cx Ans:x2 +y=k(k > 0) HR MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)